Banach Spaces and Hilbert Spaces

We begin with the following: 2.1 Definition.

A real—valued function || ‧ ||, defined on a vector space E, is called a norm if it satisfies the following conditions:

(Nl) ||x|| = 0 if and only if x = 0.

(N2) (Subadditivity) ||x + y|| ≤ ||x|| + ||y|| (for all x,y ∈ E).

(N3) (Homogeneity) ||λx|| = |λ| ||x|| (for all x ∈ E and λ ∈ K).

A vector space equipped with a norm is called a normed vector space (or normed space for short). Hereafter we shall use E (or (E,|| ‧ ||)) for a normed space, also the open unit ball in E is defined by O E = { x ∈ E : ‖ x ‖ < 1 }    ( or simply O ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq108.tif"/>

and the closed unit ball in E is defined by U E = { x ∈ E : ‖ x ‖ ≤ 1 }    ( or    simply U ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq109.tif"/>

Remark

(i). If (Nl) is replaced by the following ( N1 ) *                                         ‖ 0 ‖ = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq110.tif"/>

then the real—valued function ||x|| on E is only a seminorm, which is usually denoted by p. A vector space, equipped with a seminorm, is called a seminormed space.

Remark

(ii). It is easily seen from (N2) that any norm || ‧ || on E defines a metric d in a natural way d ( x,y ) = ‖ x − y ‖         ( for all  x , y   ∈   E ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq111.tif"/>

this metric is translation—invariant in the sense that d ( x,y ) = d ( x + z , y   + z )                     ( for all     z ∈   E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq112.tif"/>

14We shall always assume that a normed space carries this metric and its associated topology, which is called the norm—topology and denoted by || ‧ ||—top (or || ‧ ||_E—top or norm—top).

Remark

(iii). Two norms p₁ and p₂ on E are said to be equivalent if they define the same norm—topology on E. As a consequence of a local base (see (2.a.l) of (2.a)), it follows that two norms p₁ and P₂ on E are equivalent if and only if there exist λ > 0 and μ > 0 such that (2.1.a) μ ≤ p 1 ( x ) p 2 ( x ) ≤ λ                         ( for     all     0 ≠   x   ∈   E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq113.tif"/>

Remark

(iv). We say that two normed spaces (E,p) and (F,q) over K are :

metrically isomorphic (or isometric), denoted by (E, p) ≡ (F, q), if there is a bijective linear map T : E ⟶ F such that q ( Tx ) = p ( x )        ( for   all   x   ∈   E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq114.tif"/> (T is referred to as a metric isomorphism (or isometry));

topologically isomorphic (or isomorphic), denoted by (E, p) ≅ (F, q), if there is a bijective linear map T : E ⟶ F which is a homeomorphism for the norm—top (T is referred to as a topological isomorphism (or an isomorphism)).

It is easily seen that a bijective linear map T : E ⟶ F is a topological isomorphism if and only if the norm || ‧ ||_F on F, defined by   ‖ Tx ‖ F = p ( x )        ( for     all   x ∈   E ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq115.tif"/>

is equivalent to q, and this is the case (by (2.l.a)) if and only if there exist λ > 0 and μ > 0 such that μ p ( x ) ≤ q ( Tx ) ≤ λp ( x )        ( for     all  x ∈   E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq116.tif"/>

2.a <target id="page_15" target-type="page">15</target>The norm-topology:

Let (E,||‧||) be a normed space. Then the family (2.a.1) { 1 n O E : n ≥ 1 }       ( or     { 1 n U E : n ≥ 1 } ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq117.tif"/>

consisting of absolutely convex and absorbing sets, is a local base at 0 for ||‧||_E—top; moreover, the ||‧||_E—top has the following remarkable properties:

The ||‧||_E—top is compatible with the vector space operation, i.e., the maps ( x,y ) →   x + y : E   ×   E →   E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq118.tif"/>

and ( λ,x ) →   λx : K     ×   E →   E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq119.tif"/>

are continuous. (Of course, we consider the product topology on the product spaces.)

For any x₀ ∈ E and 0 ≠ γ ₀ ∈ K, the translation, defined by y →   x 0 + γ 0 y       ( for   all     y   ∈   E ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq120.tif"/>

is a homeomorphism from E onto E; consequently, the family (2.a.2) { x 0 + 1 n U E : n ≥ 1 }                   ( or  { x 0 + 1 n O E : n ≥ 1 } ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq121.tif"/>

is a local base at x₀ for the ||‧||_E—top.

As the translation and multiplication by non—zero scalars are homeomorphisms (see (2.a) (ii)), it follows that x + A ¯ = x + A ¯   ,   λA ¯ = λ A ¯         and         A ¯ + B ¯ ⊆ ( A + B ¯ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq122.tif"/>

whenever x ∈ E, A ∈ K and A, B ⊂ E (where A ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq123.tif"/> is the closure of A); moreover, we have the following:

2.2 Proposition.

Let E be a normed space and A, B ⊂ E.

16 A ¯ = ∩ n = 1 ∞ ( A + 1 n O E ) =     ∩ n = 1 ∞   ( A + 1 n U E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq124.tif"/>

A + G is open whenever G is open, hence A + IntB     ⊂     Int ( A + B )                 ( where   Int     B     denote   the     interior     of   B ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq125.tif"/>

Let K ⊂ E be compact and let B be closed. If K ⋂ B = ϕ, then there exists some m ≥ 1 such that (2.2.1) ( K + 1 m U E ) ∩ ( B + 1 m U E ) = ϕ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq126.tif"/>

Consequently, if C ⊂ E is compact and B is closed, then C + B is closed in E.

Proof.

(a) As 1 n + 1 U E ⊂ 1 n O E ⊂ 1 n U E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq127.tif"/> it follows that ∩ n = 1 ∞ ( A + 1 n O E ) =   ∩ n = 1 ∞ ( A + 1 n U E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq128.tif"/>

Now part (a) follows from the following computation: x ∈ A ¯       ⇔ ( x + 1 n O E ) ∩ A ≠ ϕ                 ( for     all     n ≥ 1 )                               ⇔ for     any     n ≥   1 ,       there     is some     a n     ∈     A     such     that                                                       x     ∈     a n +   1 n     O E             ( since     O E = −   O E   )                               ⇔ x ∈     A     + 1 n O E                 ( for     all     n ≥ 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq129.tif"/>

(b) For any X₀ ∈ E, the translation y ⟶ X₀ + y (y ∈ E) is a homeomorphism, it follows that x₀ + G is open, and hence from A + G = ∪ a ∈ A ( a + G ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq130.tif"/> that A + G is open.

Finally, since A + Int B is an open subset of A + B, it follows that A + Int B ⊂ Int (A + B).

17(c) For any x ∈ K, the closedness of B and B ⋂ K = ϕ ensure that there exists some integer n(x) ≥ 1 such that ( x + 1 n ( x ) U E + 1 n ( x ) U E + 1 n ( x ) U E ) ∩ B = ϕ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq131.tif"/>

and surely (2.2.2) ( x + 1 n ( x ) U E + 1 n ( x ) U E ) ∩ ( B + 1 n ( x ) U E ) = ϕ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq132.tif"/>

Clearly { x + 1 n ( x ) U E : x ∈ K } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq133.tif"/> ( or more precisely  { x + 1 n ( x ) O E : x ∈ K } ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq134.tif"/> forms an open covering of K, hence the compactness of K ensures that there is { x₁,…,x_k } ⊂ K such that (2.2.3) K ⊂ ∪ i=1 k ( x i + 1 n ( x i ) U E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq135.tif"/>

Let m = max{ n(x₁),…,n(x_k) }. Then 1 m ≤ 1 n ( x i )       ( i=1, ⋯ ,k ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq136.tif"/> , hence we conclude from (2.2.2) and (2.2.3) that ( K + 1 m U E ) ∩ ( B + 1 m U E ) = ϕ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq137.tif"/>

Finally, if x ∉ C + B, then (x − C) ⋂ B = ϕ. As C is compact, it follows that x − C is compact, and hence from (2.2.1) that there is m ≥ 1 such that ( ( x − C ) + 1 m U E ) ∩ ( B + 1 m U E ) = ϕ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq138.tif"/> and a fortiori ( ( x − C ) + 1 m U E ) ∩ B = ϕ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq139.tif"/>

But this implies that ( x + 1 m U E ) ∩ ( B + C ) = ϕ   ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq140.tif"/>

18in other words, x is an interior point of E \ (B + C), thus B + C is closed.

A Banach space (or more briefly, B—space) is a normed space that, regarded as a metric space, is complete.

2.b

The completion and a charaterization of completeness:

Any normed space (E, ||‧||) is isometric to a dense subspace of a Banach space E ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq141.tif"/> , called the completion of (E, ||‧||), and this is unique up to a metric isomorphism. (The closed unit ball U E ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq142.tif"/> in E ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq143.tif"/> is the closure in E ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq144.tif"/> of either U_E or O_E.)

A normed space (E, ||‧||) is separable if and only if its completion E ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq145.tif"/> is separable.

A normed space (E, ||‧||) is complete if and only if for any sequence {x_n} in E with ∑ n = 1 ∞ ‖ x n ‖ < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq146.tif"/> (called the formal series ∑ n x n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq147.tif"/> is absolutely convergent) the sequence {s_n}, defined by s n = ∑ i = 1 n x i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq148.tif"/> , is convergent to z ∈ E; in this case, we have ‖ z ‖ ≤ ∑ n = 1 ∞ ‖ x n ‖ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq149.tif"/>

Let E be a normed space, let K ⊂ E be totally bounded and closed, and let B ⊂ E be complete. If K ⋂ B = ϕ, then there exists some m ≥ 1 such that ( K + 1 m U E ) ∩ ( B + 1 m U E ) = ϕ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq150.tif"/> [Compared with (2.2) (c).]

2.3 Lemma

Let E be a normed space and M a vector subspace of E. If M is dense in E, then for any z ∈ E and ϵ > 0 there exists a sequence {u_n} in M such that z = ∑ n = 1 ∞ u n             and _             ∑ n = 1 ∞ ‖ u n ‖ ≤ ( 1 + ϵ ) ‖ z ‖ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq151.tif"/>

In other words, for any z ∈ E and ϵ > 0 there exists an absolutely convergent series ∑ n u n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq152.tif"/> of u_n ∈ M (with ∑ n ‖ u n ‖ ≤ ( 1 + ϵ ) ‖ z ‖ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq153.tif"/> ) such that ∑ n = 1 ∞ u n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq154.tif"/> converges to z.

Proof.

For each n ≥ 1, there is some x_n ∈ M such that ‖ z − x n ‖ ≤ ϵ 2 n + 1 ‖ z ‖ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq155.tif"/>

19Define u 1 = x 1             and                 u n = x n − x n − 1               ( for   all     n   ≥ 2 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq156.tif"/>

Then ∑ i = 1 n u i       = x n     ,                 ‖ u 1 ‖     =   ‖ x 1 ‖ ≤   ‖ x 1 − z ‖ + ‖ z ‖ ≤ ( ϵ 4 + 1 ) ‖ z ‖ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq157.tif"/>

and ‖ u n ‖     ≤     ‖ x n − z ‖ + ‖ x n − 1 − z ‖                     ≤ ( 1 2 n + 1 + 1 2 n ) ϵ ‖ z ‖           ( for       all     n ≥ 2 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq158.tif"/>

Consequently, z = ‖   ⋅   ‖ − lim n x n = ‖   ⋅   ‖ − lim n ∑ i = 1 n u i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq159.tif"/> and ∑ n = 1 ∞ ‖ u n ‖         ≤ [ 1 + ϵ 2 2 + ∑ n = 2 ∞ ( ϵ 2 n + 1 + ϵ 2 n ) ] ‖ z ‖                                               = [ 1 + ϵ ( ∑ n = 1 ∞ 1 2 n + 1 + ∑ n = 2 ∞ 1 2 n ] ‖ z ‖                                             = ( 1 + ϵ ) ‖ z ‖ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq160.tif"/>

The concept of convergence of a series can be used to define a basis as follows:

2.4 Definition.

Let (E, ||‧||) be a normed space. A sequence {e_n} in E is called a Schauder basis for E (also we say that E has a countable basis) if for any x ∈ E there exists a unique sequence {λ_n} in K such that x = ‖   ⋅   ‖ − lim n ∑ i = 1 n λ i e i . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq161.tif"/>

2.5 <target id="page_20" target-type="page">20</target>Proposition

A normed space (E,||‧||) with a Schauder basis must be separable.

Proof.

Let {e_n} be a Schauder basis for E and C = { ∑ j = 1 n r j e j : n ≥ 1         and         r j     are     rational } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq162.tif"/>

Then C is countable as well as dense in E, hence E is separable.

Remark.

From the preceding result, it is natural to ask the following question :

(Q) Does every separable Banach space have a Schauder basis?

This is a famous question raised by Banach more than fifty years ago. Because of almost all known separable Banach spaces had been shown to possess a Schauder basis, a positive answer was expected for a long time. Grothendieck made a deep analysis of this problem; he found many equivalent formulations and consequences but no solution; he conjectured a negative answer. In 1973, Enflo [1973] succeeded in constructing a separable, reflexive Banach space which has no Schauder basis: his ingenious but highly complicated methods were simplified to some degree by Davie [1973].

By a subspace of a normed space E is meant a vector subspace M of E; while a subspace of a B—space E is meant a closed vector subspace of E.

Let E be a normed space and M a subspace of E. We denote by J M E : M →   E     ( or     simply     J M : M →   E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq163.tif"/>

the embedding map. If M is closed in E, then E/M becomes a normed space with respect 21to the quotient norm ‖ x ( M ) ‖ = inf   { ‖ x + m ‖ : M ∈ M }   , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq164.tif"/>

where x(M) = x + M denotes the equivalence class containing x. The quotient map from E onto E/M is denoted by Q M E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq165.tif"/> or simply Q_M or Q. Of course, Q M E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq166.tif"/> is continuous, moreover it is open as shown by the following:

2.6 Proposition.

Let (E, ||‧||) be a normed space, let M be a closed vector subspace of E and Q_M: E ⟶ E/M the quotient map. Then O E/M = Q M ( O E ) ⊂ Q M ( ∪ E ) ⊂ ∪ E/M , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq167.tif"/>

hence Q_M is an open operator.

Proof.

It is easily seen that Q_M(O_E)⊂O_E/M since ‖ Q M ( x ) ‖ = inf   { ‖ x + u ‖ : u ∈ M } ≤ ‖ x ‖ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq168.tif"/>

To prove the converse, i.e., O E/M ⊆ Q M ( O E )   ,         let       Q M ( x ) ∈ O E/M . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq169.tif"/> Then inf   { ‖ x + u ‖ : u ∈ M } = ‖ Q M ( x ) ‖ < 1, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq170.tif"/>

hence the definition of infimum shows that there is a u₀ ∈ M such that (2.6.2) ‖ Q M ( x ) ‖ ≤ ‖ x + u 0 ‖ < 1. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq171.tif"/>

Now let x₀ = x + u₀. Then x₀ ∈ O_E (by (2.6.2)) is such that

Q_M(x) = Q_M(x₀) ∈ Q_M(O_E) (by (2.6.2)), hence O_E/M ⊂ Q_M(O_E).

Finally, since Q_M is continuous and U E = O ¯ E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq172.tif"/> , we conclude that Q M ( O ¯ E ) ⊂ Q M ( O E ) ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq173.tif"/> , and hence that Q M ( ∪ E ) = Q M ( O ¯ E ) ⊂ Q M ( O E ) ¯ = O ¯ E/M = ∪ E/M . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq174.tif"/>

22If E is complete (resp. separable) then so is E/M as shown by the following:

2.c Quotient spaces and product spaces:

Let (E,||‧||) and (E_i,||‧||_i) (i = 1,2,…,n) be normed spaces and M a closed vector subspace of E.

If (E,|| … ||) is complete (resp. separable), then so is the quotient space E/M (equipped with the quotient norm).

Consider the cartesian product Π i = 1 n E i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq175.tif"/> , and define, for any x = ( x 1 , ⋯ , x n ) ∈ Π i = 1 n E i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq176.tif"/> , that ‖ x ‖ ℓ ∞ = sup 1 ≤ i ≤ n ‖ x i ‖ i ;       ‖ x ‖ ℓ 2 = ( ∑ i = 1 n ‖ x i ‖ i 2 ) 1 / 2 ‖ x ‖ ℓ 1 = ∑ i = 1     n     ‖ x i ‖ i . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq177.tif"/>

Then ||‧|| _{ℓ ^∞} , ||‧|| _{ℓ ²} and ||‧|| _{ℓ ¹} are norms on Π i = 1 n E i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq178.tif"/> , and these norm topologies coincide with the product topology; moreover, Π i = 1 n E i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq179.tif"/> is complete (resp. separable) (for these norms) if and only if each (E_i||‧||_i) is complete (resp. separable). (Usually, the norm ||‧|| _{ℓ ^∞} is called the product norm of ||‧||_i (i = 1,…,n), and we also write ℓ n ∞ ( E i ) = ( ∏ i = 1 n E i , ‖   ⋅   ‖ ℓ ∞ ) ;     ℓ n 2 ( E i ) = ( ∏ i=1 n E i , ‖   ⋅   ‖ ℓ 2 )       and ℓ n 1 ( E i ) = ( ∏ i = 1 n E i , ‖   ⋅   ‖ ℓ 1 ) ⋅ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq180.tif"/>

All norms on a finite-dimensional vector space are equivalent as shown by the following:

2.7 <target id="page_23" target-type="page">23</target>Theorem.

Let (E,||‧||) be an n—dimensional normed space, and let { e₁,…,e_n} be a basis of E. Then there exist numbers α and β > 0 such that (2.7.1) α ( ∑ i = 1    n | λ i | 2 ) 1 2 ≤ ‖ x ‖ ≤ β ( ∑ i = 1  n | λ i | 2 ) 1 2       ( x = ∑ i = 1  n λ i e i ∈ E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq181.tif"/>

Consequently, any two norms on a finite—dimensional vector space are equivalent.

Proof.

Recall Cauchy—Schwarz inequality in Kⁿ: (2.7.2) ∑ i = 1 n | ζ i η i | ≤ ( ∑ i = 1 n | ζ i | 2 ) 1 2 ( ∑ i = 1 n | η i | 2 ) 1 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq182.tif"/>

Now for any x = ∑ i = 1 n λ i e i ∈ E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq183.tif"/> , we have from (2.7.2) that ‖ x ‖         ≤ ∑ i = 1 n | λ i | ‖ e i ‖                   ≤ ( ∑ i = 1 n ‖ e i ‖ 2 ) 1 2 ( ∑ i = 1 n | λ i | 2 ) 1 2                 ≤ β   ( ∑ i = 1 n | λ i | 2 ) 1 2             where     β = ( ∑ i = 1 n ‖ e i ‖ 2 ) 1 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq184.tif"/>

and hence that (2.7.3) |   ‖ x ‖ − ‖ y ‖   | ≤ ‖ x − y ‖ ≤ β ( ∑ i = 1     n | λ i − ζ i | 2 ) 1 2                                                                                                                                   ( for     any     y = ∑ i = 1     n ζ i e i   ∈   E ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq185.tif"/>

(Namely, the norm ||‧|| is a continuous function on E with respect to ℓ n 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq186.tif"/> —norm.) It is easily seen that Kⁿ is a Banach space under the norm defined by (2.7.4) [ ‖ ζ i ‖ ] 2 = ( ∑ i = 1     n |   ζ i | 2 ) 1 2   ( for     all    [ ζ i ] ∈ K n ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq187.tif"/>

24Let S be the unit sphere in Kⁿ, that is S = { [ ζ i ] ∈ K n : ‖ [ ζ i ] ‖ 2 = 1 } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq188.tif"/>

and consider a function f on S, defined by (2.7.5) f ( ζ 1 , ⋯ , ζ n ) = ‖ ∑ i = 1     n ζ i e i ‖           ( for     all     [ ζ i ] ∈ S ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq189.tif"/>

As 0 ∉ S and [ζ_i] = 0 if and only if all ζ_i= 0, it follows that (2.7.6) f ( ζ 1 , ⋯ , ζ n ) > 0           ( for     all     [ ζ i ] ∈ S ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq190.tif"/>

Moreover, (2.7.3) shows that f is continuous on S[in view of (2.7.5)]. Clearly, S is bounded and closed, hence S is compact (by Heine—Borel’s theorem), thus f attains its infimum; namely, there exists an [η _i ⁽⁰⁾] ∈ S such that α = f ( η 1 ( 0 ) , ⋯ , η n ( 0 ) ) = inf { f ( ξ 1 , ⋯ ξ n ) : f ( ξ 1 , ⋯ ξ n ) ∈ S } > 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq191.tif"/>

Now, for any 0 ≠ x = ∑ i = 1     n λ i e i ∈ E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq192.tif"/> , let ξ i = λ i ( ∑ j = 1     n | λ j | 2 ) 1/2                   ( for     all     i = 1, ⋯ ,n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq193.tif"/>

Then [ξ_i] ∈ S, hence (2.7.3), (2.7.5) and (2.7.7) s f ( ξ 1 , ⋯ ξ n )                 = ‖ ∑ i = 1     n ξ i e i ‖ = ‖ ∑ i = 1     n λ i ( ∑ j = 1     n | λ j | 2 ) 1 / 2 e i ‖                                                                 = 1 ( ∑ j = 1     n | λ j | 2 ) 1 / 2 ‖ ∑ i = 1     n λ i e i ‖ ≥ α , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq194.tif"/>

it then follows from x = ∑ i = 1     n λ i e i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq195.tif"/> that ‖ x ‖ ≥ α ( ∑ i = 1     n | λ i | 2 ) 1 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq196.tif"/>

2.8 <target id="page_25" target-type="page">25</target>Corollary.

(a) Every finite—dimensional normed space is complete.

(b) Every finite—dimensional vector subspace of a. normed space is closed.

Proof.

(a) Let E be an n—dimensional normed space and let { e₁,…,e_n } be a basis of E. Then Kⁿ ≅ E under the topological isomorphism T defined by T ( [ ξ i ] ) = ∑ i = 1     n ξ i e i             ( for   all       [ ξ i ] ∈ K n ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq197.tif"/>

As Kⁿ is complete, it follows from Kⁿ ≅ E that E is complete.

(b) Let G be a normed space and let M be an n—dimensional vector subspace of G. Then M is an n—dimensional normed space under the relative norm, hence M is complete by (a), thus M is closed in G (since the norm—topology is Hausdorff).

As a consequence of (2.8), we obtain the following interesting result.

2.9 Corollary.

If E is an infinite-dimensional Banach space, then any Hamel basis of E is uncountable. (A subset B of E is a Hamel basis if B is linearly independent and E = < B >.)

Proof.

suppose that E has a countable Hamel basis { e₁,e₂, … }. For any n ≥ 1, let M n = span     ( { e 1 , ⋯ , e n } ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq198.tif"/>

Then M_n is closed (since dim M_n = n) and E = ∪ n M n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq199.tif"/> . By Baire’s Category theorem, there is some M_K containing a non-empty open set; in particular, N(x,r) ⊂ M_K for some x ∈ M_K and r > 0. Consequently, rO E   ⊂ M K https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq200.tif"/> (since N(x,r) = x + rO_E and −x ∈ M_K), it then follows from E = ∪ n nO E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq201.tif"/> that E ⊂ M_K, and hence that dim E ≤ k, which gives a contradiction.

In order to give a characterization of finite—dimensional normed spaces, we need the 26following crucial lemma:

2.10 Riesz′s Lemma.

Let E be a normed space and M a proper, closed vector subspace of E (i.e., M ¯ = M ≠ E https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq202.tif"/> ). For any ϵ ∈ (0,1) there exists an x _ϵ ∈ E such that (2.10.1) ‖ x ϵ ‖ = 1         and _        dist     ( x ϵ , M ) = inf     { ‖ x ϵ − m ‖ : m ∈ M } ≥ ϵ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq203.tif"/>

Proof.

It is clear that ||Q_M(u)|| = dist(u,M) (for any u ∈ E). Now choose an z ∈ E\M; since Q_M(z) ≠ O(M) and the quotient norm ||‧|| is a norm, it follows that (2.10.2) ‖ Q M ( z ) ‖   =   dist   ( z   ,   M ) = α > 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq204.tif"/>

and hence from α < α ϵ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq205.tif"/> and the definition of infimum that there is some y₀ ∈ M such that (2.10.3) α ≤ ‖ z − y 0 ‖ < α ϵ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq206.tif"/>

Let us define x ϵ = z − y 0 ‖ z − y 0 ‖ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq207.tif"/>

Then ||x _ϵ || = 1; moreover, we have, for any u ∈ M, that ‖ x ϵ − u ‖               = ‖ z − y 0 ‖ z − y 0 ‖ − u ‖                                           = 1 ‖ z − y 0 ‖ ‖ z − y 0 − ‖ z − y 0 ‖ u ‖                                           = 1 ‖ z − y 0 ‖ ‖ z − ( y 0 + ‖ z − y 0 ‖ u ) ‖                                           ≥ ϵ α α = ϵ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq208.tif"/> (by (2.10.2) and (2.10.3)) since y₀ + ||z − y₀|| u ∈ M, thus dist(x _ϵ ,M) ≥ ϵ.

<target id="page_27" target-type="page">27</target>Remark.

In Riesz’s lemma, the proper vector subspace M has to be necessarily closed. For instance, let E = C[0,1] be equipped with the sup—norm ||‧||_∞ (for definition, see (2.17) (f) below) and let M be the vector subspace of all polynomials on [0,1]. Then M ¯ = C [ 0 , 1 ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq209.tif"/> and therefore the result fails to work in this case.

Also, one cannot generally take ϵ = 1 in Riesz’s lemma; in fact, the existence of u in a B—space E such that ||u|| = 1 and dist(u,M) ≥ 1 is a characterization of reflexivity of E (see (3.d) the next section or Diestel [1984, p.5—6]).

2.d Finite—dimensional Banach spaces:

Let E be a finite—dimensional normed space and M a proper, closed vector subspace of E. Then there exists a u ∈ M such that ‖ u ‖ = 1             and               dist     ( u   ,   M )   = 1. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq210.tif"/>

[Using Riesz’s lemma and Heine—Borel’s theorem on finite—dimensional space.]

2.11 Theorem (Riesz).

A normed space E is finite—dimensional if and only if its closed unit ball U_E is compact.

Proof.

Necessity. Follows from Heine—Borel’s theorem.

Sufficiency. We assume that U_E is compact, but dim E = ∞, and then show that this leads to a contradiction.

Choose x₁ ∈ E with ||x₁|| = 1; then M₁ = < {x₁} > is an 1—dimensional subspace of E which is closed and proper, by Riesz’s lemma, there is some x₂ ∈ E such that ‖ x 2 ‖ = 1                 and                 ‖ x 2 − x 1 ‖     ≥     1 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq211.tif"/>

M₂ = < {x₁,x₂} > is a 2—dimensional proper closed subspace of E, by Riesz’s lemma, there is an x₃ ∈ E such that ‖ x 3 ‖ = 1                 and                 dist     ( x 3 , M 2 ) ≥ 1 / 2 ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq212.tif"/>

28in particular, ‖ x 3 − x 2 ‖ ≥ 1 / 2             and               ‖ x 3 − x 1 ‖ ≥ 1 / 2   . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq213.tif"/>

Continue this process, we obtain a sequence {x_n} in E such that ‖ x n ‖   = 1           and             ‖ x n − x m ‖ ≥ 1 / 2           where       n   ≠ m . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq214.tif"/>

Obviously {x_n} cannot have any convergent subsequence. This contradicts the compactness of U_E, hence dim E < ∞.

An inner product space is a complex vector space H together with a function [‧,‧] : H × H ⟶ ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq215.tif"/> (called the inner product) satisfying the following

(I1) [ λ 1 x 1 + λ 2 x 2 , y ]     =   λ 1 [ x 1 , y ] + λ 2 [ x 2 ,y ]     for   all     x 1 , x 2 , y     ∈   H         and       λ 1 , λ 2 ∈ ℂ, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq216.tif"/>

(I2) [ x,y ]   =   [ y,x ¯ ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq217.tif"/> (the bar denotes the complex conjugate),

(I3) [x,x] ≥ 0 for all x ∈ H,

(14) [x,x] = 0 if and only if x = 0.

For a fixed y ∈ H, (I1) says that [‧,y] is a linear functional on H, while for a fixed x ∈ H, (I1) and (I2) indicate that [x,‧] is a conjugate—linear functional on H in the following sense [ x , μ 1 y 1 + μ 2 y 2 ] = μ ¯ 1 [ x , y 1 ] + μ ¯ 2 [ x , y 2 ] . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq218.tif"/>

Furthermore, if we define (2.12.a) ‖ x ‖ = [ x,x ] 1 2           for     all     x     ∈     H, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq219.tif"/>

29then ||x|| ≥ 0, ||x|| = 0 if and only if x = 0, and ‖ λx ‖   =   | λ | ‖ x ‖                         for     all     λ     ∈   ℂ             and     x ∈   H . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq220.tif"/>

Therefore it is natural to ask whether ||.|| is a norm. The answer is affirmative as shown by the following result.

2.12 Lemma.

Let H be an inner product space. Then (1) | [ x,y ] |   ≤   ‖ x ‖     ‖ y ‖                   for   all _       x,y   ∈   H, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq221.tif"/>

consequently, we have (2) ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖                 for   all _       x,y   ∈   H . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq222.tif"/>

Proof.

If y = 0, then (1) is trivial. Therefore we assume that y ≠ 0. For any λ ∈ ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq223.tif"/> , 0 ≤ [ x + λy,   x + λy   ]     = ‖ x ‖ 2 + λ [ y,x ] + λ ¯ [ x , y ] + λ λ ¯ ‖ y ‖ 2 ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq224.tif"/>

in particular, if λ = −[x,y]||y||⁻², the last formula is easily seen to become 0 ≤   ‖ x ‖ 2 − |   [ x , y ]   | 2 ‖ y ‖ 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq225.tif"/>

which obtains (1). To prove (2), we notice that ‖ x + y ‖ 2 = [ x + y,x + y ] = ‖ x ‖ 2 + [ x , y ] + [ y,x ] + ‖ y ‖ 2                                 =   ‖ x ‖ 2 + 2 Re [ x , y ]     + ‖ y ‖ 2 ≤ ‖ x ‖ 2 + 2 |   [ x , y ]   | + ‖ y ‖ 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq226.tif"/>

Formula (2) then follows from (1).

Formula (1) is usually called Cauchy—Schwarz’s inequality.

30Therefore every inner product space is a normed space under the norm defined by (2.12.a), which is called the associated norm.

A Hilbert space is an inner product space which is complete under the associated norm (2.12.a).

2.e Some characterizations of inner product spaces:

(a) Let H be an inner product space. Then:

(Polarization identity) For any x,y ∈ H, [ x , y ] = 1 4 { ‖ x + y ‖ 2 − ‖ x − y ‖ 2 + i ‖ x − iy ‖ 2   − i ‖ x − iy ‖ 2 } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq227.tif"/>

(Parallelogram law) For any x,y ∈ H, ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq228.tif"/>

(b) A normed space (X,||‧||) is a inner product space if and only if the norm ||‧|| satisfies the parallelogram law. [On H×H, the function, defined by [ x,y ] = 1 4 { ‖ x + y ‖ 2 − ‖ x − y ‖ 2 + i ‖ x + iy ‖ 2 − i ‖ x − iy ‖ 2 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq229.tif"/> , is an inner product.]

Let H be an inner product space. Two vectors x and y in H are said to be orthogonal, denoted by x ⊥ y, if [x,y] = 0. A subset B of H is said to be orthogonal (resp. orthonormal) if x   ⊥ y     for     all   x,y   ∈ B     ( resp . x ⊥ y     for     all  x,y   ∈ B   and   ‖ x ‖ = 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq230.tif"/>

The orthogonal complement of B, denoted by B^⊥, is defined by B ⊥ = { u ∈   H   :   [ x , u ] = 0   for   all   x ∈   B } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq231.tif"/>

2.f <target id="page_31" target-type="page">31</target>Some properties of orthogonal complements:

Let H be an inner product space and B ⊂ H.

(Pythagorean theorem) If x ⊥ y then ‖ x + y ‖ 2   = ‖ x ‖ 2 + ‖ y ‖ 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq232.tif"/>

B^⊥ is a closed vector subspace of H.

If B is an orthogonal set of non-zero vectors, then B is linearly independent.

Let Λ be a non-empty set. Then the family F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq233.tif"/> (Λ) of all non-empty finite subsets of Λ is a directed set under the set inclusion. If [x_i,Λ] is a family in H, for any α ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq234.tif"/> (Λ) we write x α = ∑ i ∈ α x i , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq235.tif"/>

hence {x _α ,α ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq236.tif"/> (Λ)} becomes a net in H which is called the associated net with [x_iΛ]. If {x _α ,α ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq237.tif"/> (Λ)} is convergent under the norm, then we write ∑ Λ x i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq238.tif"/> to be its limit, that is ∑ Λ x i = lim α ∈ F ( Λ ) ∑ i ∈ α x i . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq239.tif"/>

The following result is a generalization of the Pythagorean theorem.

2.13 Proposition.

Let H be a Hilbert space and [x_i,Λ] an orthogonal family in H. Then the following statements are equivalent.

The net {x _α , α ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq240.tif"/> (Λ)} associated with [x_i,Λ] is convergent.

∑ Λ ‖ x i ‖ 2 < ∞ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq241.tif"/>

In this case, we have ‖ ∑ Λ x i ‖ 2 = ∑ Λ ‖ x i ‖ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq242.tif"/>

<target id="page_32" target-type="page">32</target>Proof.

(a) ⇒ (b) : In view of the continuity of the norm and the Pythagorean theorem, it follows from the existence of ∑ Λ x i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq243.tif"/> that ‖ ∑ Λ x i ‖ 2 = ‖     lim α ∈ F ( Λ )     ∑ i ∈ α x i ‖ 2 = lim α ∈ F ( Λ ) ‖   ∑ i ∈ α x i ‖ 2                                       =   lim α ∈ F ( Λ ) ∑ i ∈ α ‖ x i ‖ 2 =   < ∞ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq244.tif"/>

and hence that ‖ ∑ Λ x i ‖ 2 = ∑ Λ ‖ x i ‖ 2 < ∞   . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq245.tif"/>

(b) ⇒ (a) : For any ϵ > 0 there exists an α _∘ ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq246.tif"/> (Λ) such that ∑ i ∈ α ‖ x i ‖ 2 − ∑ i ∈ α Ο ‖ x i ‖ 2 < ϵ   for   all     α ≥ α Ο . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq247.tif"/>

If α ₁, α ₂ ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq248.tif"/> (Λ) are such that α _j ≥ α _o (j = 1,2), then we have by the Pythagorean theorem that ‖   ∑ i ∈ α 1 x i − ∑ i ∈ α 2 x i ‖ 2 = ∑ i ∈ α 1 \ α 2 ‖ x i ‖ 2 + ∑ i ∈ α 2 \ α 1 ‖ x i ‖ 2                             ≤ ∑ i ∈ α 1 ∪ α 2 ‖ x i ‖ 2 − ∑ i ∈ α Ο ‖ x i ‖ 2 ≤ ϵ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq249.tif"/>

hence { ∑ i ∈ α x i , α ∈ F ( Λ ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq250.tif"/> is a Cauchy net in H, thus the completeness of H ensures that ∑ Λ x i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq251.tif"/> exists.

2.14 Corollary.

Let H be a Hilbert space, let [e_i,Λ] be an orthonormal family in H and let M be the smallest closed vector subspace of H generated by the set {e_i : i∈Λ}. Then M = { ∑ Λ λ i e i : λ i ∈ ℂ             and _             ∑ Λ | λ i | 2 < ∞ } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq252.tif"/>

Proof.

We first notice that if ∑ Λ | λ i | 2 < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq253.tif"/> then the set {λ _i e _i .: i ∈ Λ} is orthogonal and lim α ∈ F ( Λ ) ∑ i ∈ α ‖ λ i e i ‖ 2 = lim α ∈ F ( Λ ) ∑ i ∈ α | λ i | 2 = ∑ Λ | λ i | 2 =   < ∞ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq254.tif"/>

hence lim α ∈ F ( Λ ) ∑ i ∈ α λ i e i     = ∑ Λ λ i e i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq255.tif"/> exists in H by (2.13). Thus the set N = { ∑ Λ λ i e i : λ i ∈ ℂ ,   ∑ Λ | λ i | 2 < ∞ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq256.tif"/>

33is a subset of H containing all e_i (i ∈ Λ). Further if ∑ Λ λ i e i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq257.tif"/> and ∑ Λ μ i e i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq258.tif"/> belong to N, then ∑ Λ | λ i + μ i | 2 ≤ 2 ( ∑ Λ | λ i | 2 + ∑ Λ | μ i | 2 )   < ∞ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq259.tif"/>

hence ∑ Λ λ i e i + ∑ Λ μ i e i ∈ N . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq260.tif"/> Consequently N is a vector subspace of H. This show that N ⊂ M.

In order to verify this result, it suffices to show that N is closed. Indeed, let {x^(m)} be a Cauchy sequence in N and x ( n ) = ∑ Λ λ i ( n ) e i                 ( for     all     n ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq261.tif"/>

For any i ∈ Λ, | λ i ( n ) − λ i ( m ) | ≤ ( ∑ Λ | λ i ( n ) − λ i ( m ) | 2 ) 1 2 = ‖ x ( n ) − x ( m ) ‖ ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq262.tif"/>

this implies that { λ i ( m ) , m ≥ 1 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq263.tif"/> is a Cauchy sequence in ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq264.tif"/> , thus for any i ∈ Λ, we have λ i = lim m λ i ( m ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq265.tif"/> . We claim that ∑ Λ | λ i | 2 < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq266.tif"/> . Indeed, for any α ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq267.tif"/> (Λ), Σ i ∈ α | λ i | 2 lim n Σ i ∈ α | λ i ( n ) | 2 ≤ lim n Σ Λ | λ i ( n ) | 2 = lim n ‖ x ( n ) ‖ 2 < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq268.tif"/>

in view of the completeness of H. Therefore x = ∑ Λ λ i e i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq269.tif"/> is well-defined and belongs to N. Now for any ϵ > 0, we choose n₀ > 0 satisfying ‖ x ( n ) − x ( m ) ‖ < ϵ                               ( for   all     n,     m     ≥     n 0 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq270.tif"/>

For any α ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq271.tif"/> (Λ), Σ i ∈ α | λ i − λ i ( n ) | 2 lim n Σ i ∈ α | λ i ( m ) − λ i ( n ) | 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq272.tif"/> ≤ lim n     sup   ‖ x ( m ) − x ( n ) ‖ 2 ≤ ϵ 2               whenever     n     ≥   n ∘ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq273.tif"/>

34Thus for any n > n_o, we have ‖ x − x ( n ) ‖ 2 = Σ Λ | λ i − λ i ( n ) | 2 ≤ ϵ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq274.tif"/>

which obtains our assertion.

A family [e_i,Λ] in a Hilbert space H is called an orthonormal basis if the set {e_i: i∈Λ} is orthonormal and the smallest closed vector subspace containing all e_i (i∈Λ) is H.

2.15 Corollary.

If {e_i i∈Λ} is an orthonormal basis for a Hilbert space H, then for any x ∈ H there exists uniquely a family {λ _i : i ∈ Λ} (called the Fourier coefficients contained in ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq275.tif"/> such that x = ∑ Λ λ i e i ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq276.tif"/> moreover λ i = [ x,e i ]             for     all _     i ∈ Λ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq277.tif"/>

Proof.

By (2.14), there exist uniquely {λ_i : i ∈ Λ} with ∑ Λ | λ i | 2 < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq278.tif"/> such that x = ∑ Λ λ i e i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq279.tif"/> . For any i∈Λ, the continuity of the inner product ensures that [ x , e i ] = [ lim α ∈ F ( Λ ) Σ j ∈ α λ j e j ,e i ] = lim α ∈ F ( Λ ) Σ j ∈ α λ j [ e j ,e i ] = λ i . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq280.tif"/>

2.16 Theorem.

Every Hilbert space H possesses an orthonormal basis.

Proof.

Let M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq281.tif"/> be the collection of all orthonormal subsets B of H with the partial ordering B₁ ≤ B₂. Let {B_j : j ∈ D} be a chain in M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq282.tif"/> . Then ∪ j ∈ D B j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq283.tif"/> is obviously orthonormal, hence it is an upper bound of {B_j : j ∈ D}. Zorn’s lemma ensures that M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq284.tif"/> has a maximal element B. To show that B is an orthonormal basis for H, let M be the smallest closed 35subspace of H containing B. If M ≠ H, then there exists an e ∈ M^⊥ with ||e|| = 1, hence the set B ⋃ {e} is an orthonormal subset of H containing B properly. This contradicts the maximality of B.

We conclude this section with some examples of Banach spaces.

2.17 Examples.

(a) The vector space, defined by ℓ ∞ = { ξ = [ ξ n ] ∈ K N : [ ξ n ]    is     bounded } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq285.tif"/>

is a Banach space under the sup—norm ||‧||_∞, defined by ‖ ξ ‖ ∞ = sup j ≥ 1 | ξ j |       ( for   all     ξ = [ ξ n ] ∈ ℓ ∞ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq286.tif"/>

But ℓ ^∞ is not separable.

Proof.

It is not hard to show that ℓ ^∞ is a Banach space. To see that ℓ ^∞ is not separable, let B = { ξ = [ ξ j ] ∈ ℓ ∞ : ξ j = 1         or         ξ j = 0     for     all     j ≥ 1 } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq287.tif"/>

Then it is easily seen that

B = {0,1} ^{ℕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq288.tif"/>} (the set of all maps from ℕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq289.tif"/> into the 2—point set {0,1 })

and that ‖ ξ − η ‖ ∞ = 1           for   all         ξ ,     η ∈ B           with           ξ ≠ η   , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq290.tif"/>

hence B is not countable since card B = 2^{card ℕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq291.tif"/>} = 2 ^{N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq292.tif"/> ₀} = c (the power of the continum).

36It then follows that B cannot contain a countable dense subset; in other words, B is not separable, thus ℓ ^∞ is not separable. (Since it is easily shown that any subset of a separable normed space must be separable.)

(b) The subsets, defined by c = { ξ = [ ξ i ] ∈ ℓ ∞ : lim j     ξ j     exists } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq293.tif"/>

and c 0 = { ξ = [ ξ i ] ∈ ℓ ∞ : lim j     ξ j = 0 } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq294.tif"/>

are vector subspaces of ℓ ^∞. Equipped with the sup—norm ||‧||_∞, c and c₀ are separable Banach spaces. Furthermore, for any n ≥ 1, let e ( n ) = [ δ j ( n ) ] j ≥ 1 and    e = ( 1 , 1 , … ) ,                                         where     δ j ( n )   = { 1 if j = n 0 if j ≠ n  . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq295.tif"/>

Then the countable set {e⁽ⁿ⁾:n≥l}⋃{e} is a Schauder basis for c, while { e⁽ⁿ⁾ : n ≥ 1 } is a Schauder basis for c₀.

Remark

The space c contains c₀ as a subspace, and c₀ has codimension one, thus c = c 0 ⊕ < { e } > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq296.tif"/>

Every convergent sequence [ξ_n] ∈ c can be represented in the form [ ξ n ] = [ η n ] + λe           with             λ = lim j   ξ j         and          [ ξ n − λ ] ∈ c 0 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq297.tif"/>

37(As lim j ξ j = λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq298.tif"/> , it follows that lim n     sup j ≥ n | ξ j − λ | = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq299.tif"/> , and hence that     ‖ ∑ k = 1 ∞ ξ k e ( k ) − ∑ j = 1 n ( ξ j − λ ) e ( j ) − λe ‖ ∞ =   ‖ ( λ, … ,λ, ξ n + 1 ⋯ ) − λe ‖ ∞ = sup j ≥ n + 1       | ξ j − λ | → ​   0     ( as     n   → ​   ∞ ) . ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq300.tif"/>

(c) The vector space, defined by ℓ 1 = { ξ = [ ξ n ] ∈ K ℕ : Σ n | ξ n | < ∞ } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq301.tif"/>

is a separable Banach space under the norm ‖ [ ξ n ] ‖ ℓ 1 = Σ n = 1 ∞ | ξ n |             ( for     all     [ ξ n ] ∈ ℓ 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq302.tif"/>

Furthermore, the sequence { e⁽ⁿ⁾, n ≥ 1 } is a Schauder basis for ℓ ¹.

(d) For any p with 1 ≤ p > ∞, the set, defined by ℓ p = { ξ = [ ξ j ] ∈ K ℕ : Σ j = 1 ∞ | ξ j | p < ∞ } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq303.tif"/>

is a separable Banach space under the norm ‖ ξ ‖ p = ( ∑ j = 1   ∞ | ξ k | p ) 1 /p       ( for     all     ξ = [ ξ j ] ∈ ℓ p ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq304.tif"/>

Moreover, the sequence { e⁽ⁿ⁾, n ≥ 1 } is a Schauder basis for ℓ ^p

(e) If     1 ≤ p 1 < p 2 ≤ ∞ ,     then     ℓ p 1 ⊊ ℓ p 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq305.tif"/> (e.1) ‖ [ ξ j ] ‖ p 2               = ( ∑ k = 1   ∞ | ξ k | p 2 ) 1 / p 2 ≤ ( ∑ k = 1   ∞ | ξ k | p 1 ) 1 /p 1                                         = ‖ [ ξ j ] ‖ p 1           ( for     all     [ ξ i ] ∈ ℓ p 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq306.tif"/> (e.2) lim p → ∞ ‖ [ η j ] ‖ p             = lim p → ∞ ( ∑ k = 1   ∞ | η k | p ) 1 / p = inf p ≥ 1   ‖ [ η j ] ‖ p                                                     = sup j ≥ 1 | η j | = ‖ [ η j ] ‖ ∞       ( for     all       [ η j ] ∈ ℓ p     ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq307.tif"/>

<target id="page_38" target-type="page">38</target>Proof.

To prove the first part, let 0 ≠ [ζ_i] ∈ ℓ ^p₁ and (1) η j = ζ j ‖     [ ζ i ]     ‖ p     ( for     all     j = 1 , 2 , .. ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq308.tif"/>

It then follows that |η _j| ≤ 1 (for all j ≥ 1), and hence from 1 ≤ p₁ < p₂ ≤ ∞ that | η j | p 2 ≤ | η j | p 1               ( for     all     j ≥ 1 ) ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq309.tif"/>

thus ∑ j = 1 ∞ | η j | p 2 ≤ ∑ j = 1 ∞ | η j | p 1 = 1. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq310.tif"/>

Consequently, (2) ( ∑ j = 1 ∞ | ζ j | p 2 ) 1/p 2 ≤ ‖     [ ζ j ]     ‖ p 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq311.tif"/> (by (1)); this implies that [ζ _j] ∈ ℓ ^p₂ and that (e.l) holds.

To prove (e.2), we first notice from (e.l) that ( ∑ k = 1 ∞ | η k | p ) 1/p ↓    ( for     all     p ≥ 1 ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq312.tif"/> hence (3) lim p → ∞ ( ∑ k = 1 ∞ | η k | p ) 1 /p = inf p ≥ 1 ( ∑ k = 1 ∞ | η k | p ) 1 /p . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq313.tif"/>

39On the other hand, for any ϵ > 0, p₀ ≥ 1 and [η _i] ∈ ℓ ^p₀ , there is some N ≥ 1 such that ( ∑ k>n ∞ | η k | p 0 ) 1 /p 0 < ϵ                   ( for     all     n ≥ N ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq314.tif"/>

For all p with p > p₀, we obtain from (e.l) that ( ∑ k = 1 ∞ | η k | p ) 1 /p             ≤ ( ∑ k = 1 N | η k | p ) 1 /p + ( ∑ k = N+1     ∞ | η k | p ) 1 /p                                                                       ≤ ( ∑ k = 1 N | η k | p ) 1 /p + ( ∑   k = N+1     ∞ | η k | p 0 ) 1 /p 0                                                                     ≤ ( ∑   k = 1 N | η k | p ) 1 /p + ϵ                                                                     ≤ N 1 / p max 1 ≤ k ≤ N         | η k | + ϵ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq315.tif"/> (since ∑ k = 1 N | η k | p ≤ N   max 1 ≤ k ≤ N | η k | p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq316.tif"/> ), so that (4) lim p → ∞ ( ∑ k = 1 ∞ | η k | p ) 1 /p ≤ max 1 ≤ k ≤ N   | η k | + ϵ ≤ sup j ≥ 1 | η j | + ϵ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq317.tif"/>

By the definition of supremum, there is some η _k0 such that | η k 0 | > sup j ≥ 1 | η j | − ϵ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq318.tif"/>

Clearly, | η k 0 | p ≤ ∑ k = 1 ∞ | η k | p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq319.tif"/> , it then follows that ( ∑ k = 1 ∞ | η k | p ) 1 / p ≥ | η k 0 | > sup j ≥ 1 | η j | − ϵ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq320.tif"/>

40and hence from (3) that (5) lim p → ∞ ( ∑   k = 1 ∞ | η k | p ) 1 / p ≥ sup j ≥ 1 | η j | − ϵ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq321.tif"/>

Combining (4) and (5), we obtain (e.2).

Finally, to prove that ℓ ^P₁ ≠ ℓ ^p₂ , let p = p 1 + p 2 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq322.tif"/> and ξ k = k − 1 / p     ( for     all     k ≥ 1 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq323.tif"/>

Then p 1 < p < p 2 , p 2 p > 1         and           p 1 p < 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq324.tif"/> it follows that [ k − 1 / p ] k ≥ 1 ∈ ℓ p 2               but               [ k − 1 / p ] k ≥ 1 ∉ ℓ p 1 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq325.tif"/>

(f) The space C(Ω): Let Ω be a compact Hausdorff space, and let C(Ω) (resp. C _{R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq326.tif"/>} (Ω)) be the set of all complex—valued (or real—valued) continuous functions on Ω. Then C(Ω) (resp. C _{R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq327.tif"/>} (Ω)) is a Banach space under the sup—norm ||‧||_∞, defined by ‖ f ‖ ∞ = max t ∈ Ω     | f ( t ) |           ( for      all     f ∈ C ( Ω ) ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq328.tif"/>

(g) For 1 ≤ p < ∞, let L^(p)[a,b] (where −∞ < a < b < ∞) be the vector space consisting of all complex—valued (or real—valued) Lebesgue measurable functions x(‧) on [a,b], which are p—th power integrable in the sense that ‖ x ‖ p = ( ∫ a b | x ( t ) | p dm ( t ) ) 1 / p   ( for      all     x ( ⋅ ) ∈ L ( p ) [ a,b ] ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq329.tif"/> (where m always denote the Lebesgue measure on [a,b]), and let N = { x ( ⋅ ) ∈ L ( p ) [ a,b ] : x ( t ) = 0     a .e } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq330.tif"/>

41Then N is a closed vector subspace of L^(p)[a,b], hence the quotient space L^(p)[a,b]/N, equipped with the quotient norm ||‧||_p, is a Banach space, which is usually denoted by L^p[a,b].

(h) The space L^∞[a,b]: Let L^∞[a,b] be the set of all essentially bounded, Lebesgue measurable functions on [a,b], more precisely the set of equivalence classes of all essentially bounded, Lebesgue measurable functions on [a,b]. (A Lebesgue measurable function x(‧) on [a,b], except perhaps a set of measure zero, is said to be essentially bounded if there is a C ≥ 0 such that |x(‧)| ≤ C a.e. on [a,b]). Then L^∞[a,b] is a Banach space under the essential sup—norm, defined by ‖ x ‖ ∞ = ess         sup [ a,b ] | x ( ⋅ ) | = inf { sup t ∈ [ a,b ] \ E 0 | x ( t ) | : m ( E 0 ) = 0 } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq331.tif"/>

The following example should be compared with Example (e).

(i) If _   1   ≤ p 1 < p 2 < ∞ ,           then _         L p 2 [ a,b ] ⊊ L p 1 [ a,b ] , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq332.tif"/> (i.1) ‖ x ‖ p 1 ≤ ‖ x ‖ p 2 ( b − a ) 1 p 1 − 1 p 2           ( for     all     x ( ⋅ ) ∈ L p 2 [ a,b ] ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq333.tif"/>

and (i.2) ‖ x ‖ ∞ = lim p → ∞ p ≥ 1 ‖ x ‖ p           ( for     all     x ( ⋅ ) ∈ L ∞ [ a,b ] ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq334.tif"/>

Note. The preceding example holds only for a finite measure space; for a general measure space no such inclusion holds.

<target id="page_42" target-type="page">42</target>Proof.

For any x(‧) ∈ L^p₂ [a,b], we have | x ( ⋅ ) | p 1 ∈ L p 2 p 1 [ a,b ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq335.tif"/> . Now let q be the conjugate index of p 2 p 1 ( i .e ., 1 q + 1 p 2 p 1 = 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq336.tif"/> . Then q = p 2 p 2 − p 1 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq337.tif"/>

Hence we apply Hölder’s ineqality to |x(‧)|^p₁ and |y(‧)| = 1, we obtain ∫ a b | x ( t ) p 1 d m ( t ) |                       ≤ ( ∫ a b | x ( t ) | p 1 ) p 2 p 1 d m ( t ) ) p 1 p 2 ( b − a ) p 2 − p 1 p 2                                                                                     = ( ∫ a b | x ( t ) | p 2 )   d m ( t ) ) p 1 p 2 ( b − a ) p 2 − p 1 p 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq338.tif"/>

thus ‖ x ‖ ∞ = ( ∫ a b | x ( t ) | p 1 dm ( t ) ) 1 /p 1 ≤ ‖ x ‖ p 2 ( b − a ) p 2 − p 1 p 1 p 2                   = ‖ x ‖ p 2 ( b − a ) 1 p 1 − 1 p 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq339.tif"/>

This proves (i.l) and L^p₂ [a,b] ⊂ L^p₁ [a,b].

To prove (i.2), let x(‧) ∈ L^∞[a,b] and λ = ‖ x ‖ ∞ = ess     sup [ a,b ] | x ( ⋅ ) | . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq340.tif"/>

Then |x(‧)| ≤ λ on [a,b ‖ x ‖ p ≤ ∫ a     b λ P dm ( t ) ) 1 p = λ ( b − a ) 1 p . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq341.tif"/>

43The inequality (i.l) shows that ||x||_p ↑ (for all p ≥ 1), so that lim p → ∞ ¯ ‖ x ‖ p = sup p ≥ 1 ‖ x ‖ p ≤ λ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq342.tif"/>

On the other hand, for any ϵ > 0, there exists a Lebesgue measurable set B₀ in [a,b] such that m ( B 0 ) = 0             and             sup t ∈ [ a,b ] \ B 0 | x ( t ) | > λ − ϵ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq343.tif"/>

Now the set B = [a,b]\B₀ is measurable with 0 < m(B) ≤ b − a such that ‖ x ‖ p ≥ ( ∫ B | x ( t ) | p ) dm ( t ) ) 1 p > ( λ − ϵ ) ( m ( B ) ) 1 p , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq344.tif"/>

so that lim p → ∞ ‖ x ‖ p ≥ λ − ϵ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq345.tif"/>

Thus lim p → ∞ ‖ x ‖ p = λ = ‖ x ‖ ∞ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq346.tif"/>

By a similar argument give in the proof of (a)—(e) of (2.17), one can verify the following more general Banach (sequence) spaces.

2.g The spaces

ℓ ^p(Λ) (I): Let Λ be a non-empty index set. Then the collection of all non-empty finite subsets of Λ, denoted by F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq347.tif"/> (Λ), is a directed set ordered by the set inclusion. Elements in F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq348.tif"/> (Λ) will be denoted by α,β,γ etc. A family [λ_i,Λ] of numbers is said to be summable if the net { ∑ i ∈ α λ i , F ( Λ ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq349.tif"/> converges. The uniquely determined limit λ is called the sum of [λ_i,Λ], and denoted by λ= ∑ Λ λ i         or       λ = ∑ i λ i . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq350.tif"/>

44A family [λ_i,Λ] of numbers is summable if and only if it is absolutely summable in the sense that the family [|λ_i|,Λ] of positive numbers is summable, and this is the case if and only if sup α ∈ F ( Λ ) ∑ i ∈ α | λ i | < ∞ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq351.tif"/>

If [λ_i,Λ] is a summable family of numbers, then it contains at most countably many non—zero terms.

For 1 ≤ p < ∞, the collection ℓ ^p(Λ) of all families [λ_i,Λ] of numbers for which [|λ_i|^P,Λ] is summable forms a Banach space with respect to the operation (2.g.1) a [ λ i , Λ ] + b [ μ i , Λ ] = [ aλ i + b μ i , Λ ]               ( a , b ∈ K,   [ λ i ] , [ μ i ] ∈ ℓ p ( Λ ) ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq352.tif"/>

and with the norm ‖ [ λ i , Λ ] ‖ p = ( Σ i | λ i | p ) 1 p . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq353.tif"/>

In particular, if Λ = N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq354.tif"/> then ℓ ^p( N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq355.tif"/> ) is the usual Banach spaces ℓ ^p.

The collection ℓ ^∞(Λ) of all bounded families [λ_i,Λ] of numbers forms a Banach space with respect to the operation (2.g.l) and the norm (2.g.2) ‖ [ λ i , Λ ] ‖ ∞     = sup i ∈ Λ | λ i |         ( for       all     [ λ i ] ∈ ℓ ∞ ( Λ ) ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq356.tif"/>

A family [λ_i,Λ] of numbers is called a null family (or converge to 0) if for any ϵ > 0 there exists an α ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq357.tif"/> (Λ) such that | λ i | < ϵ         for     all     i ∉ α . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq358.tif"/>

45The collection C₀(Λ) of all null families of numbers forms a Banach space with respect to the operation (2.g.l) and with the sup—norm (2.g.2). In particular, if Λ = N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq359.tif"/> then ℓ ^∞( N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq360.tif"/> ) and C₀( N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq361.tif"/> ) are the usual Banach spaces ℓ ^∞ and C₀ respectively.

As usual, we denote the j—th unit family by e ( j ) = [ δ ji , i ∈ Λ ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq362.tif"/> , where δ _ji is Kronecker’s symbol. Therefore each element [λ_i,Λ] in ℓ ¹(Λ) (or in C₀(Λ)) has the following representation : (2.g.3) [ λ i , Λ ]             = Σ Λ λ j e ( j )     and https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq363.tif"/> (2.g.4) λ j =     < [ λ i , Λ ] , e ( j ) >               ( for   all   j ∈ Λ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq364.tif"/>

Of course, the convergence of (2.g.3) is understood for the ℓ ¹(Λ)—norm or the ||‧||_∞—norm dependent upon on the element [λ_i,Λ] belonging to ℓ ¹(Λ) or C₀(Λ). It should be noted that elements [λ_i,Λ] in ℓ ^∞(Λ) cannot have any representation analogues to (2.g.3), but we still have λ j =   < [ λ i , Λ ] , e ( j ) >                 ( for   all   j ∈ Λ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq365.tif"/>