Vector Topologies

A natural generalization of normed spaces is vector spaces equipped with a topology which is induced by a family of seminorms instead of one norm. Vector spaces with such a topology are special cases of a more general class of spaces, called topological vector spaces, as defined by the following: 7.1 Definition.

Let X be a vector space over K. A topology P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq901.tif"/> on X is called a vector topology if it is compatible with the algebraic operations of X in the following sense:

(VT1) the map (x,y) → x + y : X × X → X is continuous;

(VT2) the map (λ,y) → λy : K × X → X is continuous;

A vector space equipped with a vector topology is called a topological vector space (abbreviated TVS). Hereafter we shall generally denote a TVS by ( X , P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq902.tif"/> or X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq903.tif"/> or simply by X if the vector topology on X does not require any special notation; and the term local base, denoted by U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq904.tif"/> , will always mean a local base at 0 for P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq905.tif"/> .

Two TVS X and Y over K are said to be topologically isomorphic, denoted by X ≅ Y, if there exists a bijective linear map T : X → Y which is a homeomorphism; T is referred to as a topological isomorphism from X onto Y.

For a TVS X, the whole topological structure of X is determined by a local base (hence a linear map T : X → Y is continuous if and only if it is continuous at 0), as shown by the following simple, but important result which is similar to (2.a).

7.a Homeomorphism of translation:

Let X be a TVS. For any x₀ ∈ X and 0 ≠ r₀ ∈ K, the translation,defined by y → x 0 + r 0 y           ( for    all    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/eq906.tif"/>

111is a homeomorphism from X onto X.

Vector topologies can be characterized by means of a local base as shown by the following result.

7.2 Theorem

In a TVS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq907.tif"/> , there exists a local base U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq908.tif"/> whose members have the following properties:

(NSl) every member in U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq909.tif"/> is circled and absorbing;

(NS2) for anv V ∈ U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq910.tif"/> there is an W ∈ U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq911.tif"/> such that W + W ⊂ V.

Conversely, if U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq912.tif"/> is a filter base on X (i.e., any element in U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq913.tif"/> is non-empty and for any V 1 ,  V 2 ∈ U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq914.tif"/> there is an V ∈ U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq915.tif"/> with V ⊂ V₁ ∩ V₂ which satisfies (NSl) and (NS2), then there exists a unique vector topology T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq916.tif"/> on X such that U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq917.tif"/> is a local base at 0 for T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq918.tif"/> .

Proof.

The continuity of the map (λ,x) → λx at (0,0) ensures that the family of all circled 0–neighbourhoods in X is a local base at 0. [For any 0–neighbourhood W in X there is a 0–neighbourhood U and δ > 0 such that λU ⊂ W (for all | λ | ≤ δ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq919.tif"/> ), hence the set V = ∪ {  λU :  | λ | ≤ δ   } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq920.tif"/> is a circled 0–neighbourhood with V ⊂ W.] For any fixed x₀ ∈ X, the continuity of (λ,x₀) → λx₀ at λ = 0 ensures that every 0–neighbourhood (in particular, every V ∈ U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq921.tif"/> ) is absorbing.

By (VT1),the continuity of the map (x,y) → x + y at (0,0) ensures that U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq922.tif"/> satisfies the condition (NS2).

Conversely, we first define a topology T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq923.tif"/> on X by setting J = { ϕ ≠ G ⊂ X : x ∈ G    implies    x + V ⊂ G     ( for    some   V ∈ U ) }  U { ϕ } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq924.tif"/>

Clearly T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq925.tif"/> is the unique topology on X such that U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq926.tif"/> is a local base at 0 for T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq927.tif"/> . We complete the proof by showing that T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq928.tif"/> is a vector topology.

112By (NS2), the continuity of the map (x,y) → x + y at (x₀,y₀) follows from ( x 0 + W ) + ( y 0 + W ) ⊂ x 0 + y 0 + V https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq929.tif"/> .

To prove the continuity of the map (λ,y) → λy at (λ₀,y₀), let λ₀y₀ + U be any λ₀y₀–neighbourhood, where U ∈ U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq930.tif"/> . Suppose that n is a natural number with | λ 0 | ≤ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq931.tif"/> . Then (NS2) implies that there is an V ∈ U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq932.tif"/> such that V + ⋯ + V https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq933.tif"/> (n+2 summands) is contained in U; consequently, nV + V + V ⊂ U (since nV ⊂ V + ⋯ + V https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq934.tif"/> (n summands)). As V is absorbing, there is a natural number m ≥ 1 such that (1) y 0 ∈ mV . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq935.tif"/>

Suppose now that λ ∈ K is such that | λ − λ 0 | ≤ 1 m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq936.tif"/> and that y ∈ y₀ + V. We claim that λy ∈ λ o y 0 + U, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq937.tif"/>

hence the assertion follows. Indeed, since V is circled, we have (2) ( λ − λ 0 ) y 0 ∈ ( λ − λ 0 ) mV ⊂ V https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq938.tif"/> (by(1)), (3) ( λ − λ 0 ) ( y − y 0 ) ∈ ( λ − λ 0 ) V ⊂ 1 m V ⊂ V, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq939.tif"/> (4) λ 0 ( y − y 0 ) ∈ λ 0 V ⊂ nV https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq940.tif"/> (since | λ 0 | ≤ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq941.tif"/> ); it then follows from (2),(3) and (4) that λy = λ 0 y 0 + ( λ − λ 0 ) y 0 + ( λ − λ 0 ) ( y − y 0 ) + λ 0 ( y − y 0 ) ∈ λ 0 y 0 + V + V + nV ⊆ λ 0 y 0 + U ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq942.tif"/>

this proves our assertion.

Remark

As a consequence of (7.2), we see that every 0–neighbourhood in X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq943.tif"/> contains a closed 0–neighbourhood; hence there exists a local base U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq944.tif"/> at 0 consisting of closed, circled and absorbing sets.

7.3 Definition.

A vector topology P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq945.tif"/> on X is said to be locally convex if P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq946.tif"/> 113admits a local base at 0 consisting of convex sets. A vector space equipped with a Hausdorff locally convex topology is called a locally convex space (abbreviated LCS).

Locally convex topologies can be characterized by means of local base as shown by the following result.

7.4 Theorem

Let P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq947.tif"/> be a locally convex topology on X. Then there exists a local base U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq948.tif"/> whose members have the following properties:

(LC1) every member in U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq949.tif"/> is absolutely convex and absorbing;

(LC2) if V ∈ U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq950.tif"/> and λ > 0 then λV ∈ U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq951.tif"/> .

Conversely, if U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq952.tif"/> is a filter base on X which satisfies conditions (LC1) and (LC2), then there exists a unique locally convex topology T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq953.tif"/> on X such that U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq954.tif"/> is a local base at 0 for T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq955.tif"/> .

Proof.

There exists, by (7.3), a local base M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq956.tif"/> consisting of convex 0–neighbourhoods. If V ∈ M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq957.tif"/> , by (7.2) there is a circled 0–neighbourhood W such that W ⊂ V, hence W ⊂ co W ⊂ V, thus co W is a convex 0–neighbourhood which is obviously circled; consequently, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq958.tif"/> admits a local base V https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq959.tif"/> consisting of absolutely convex 0–neighbourhoods. Now the family, defined by U X = {  λW : W ∈ V   , 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/eq960.tif"/>

has the required properties.

Conversely, the assertion follows from (7.2) and (7.3).

Locally convex topologies can be determined by a family of seminorms. To do this, we require the following terminology and result (see (7.b) below). A family ℙ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq961.tif"/> of seminorms on X is saturated if max 1 ≤ i ≤ n  p i ∈ ℙ    whenever    p i ∈ ℙ ( i = 1 , 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/eq962.tif"/>

114where ( max 1 ≤ i ≤ n  p i ) x = max 1 ≤ i ≤ n  p i ( x )      ( for all x  ∈  X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq963.tif"/> .

7.b The continuity of seminorms:

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq964.tif"/> be a TVS, let V be an absolutely convex, absorbing subset of X and p_V the gauge of V. Then:

(i) Int V ⊂ {  x ∈ X : p V ( x ) < 1 } ⊂ V ⊂ {  x ∈ X : p V ( x ) ≤ 1 } ⊂ V ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq965.tif"/> .

(ii) V is a P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq966.tif"/> -neighbourhood of 0 if and only if p_V is continuous; in this case, Int   V = { x ∈ X : p V ( x ) < 1 }    and   V ¯ = {  x ∈ X : p V ( 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/eq967.tif"/>

hence P_V is the gauge of Int V as well as of V ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq968.tif"/> (where Int V is the interior of V).

7.5 Theorem

A locally convex topology P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq969.tif"/> on X can always be defined by a saturated family of seminorms, e.g. by the family of all P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq970.tif"/> -continuous seminorms on X. Conversely, if ℙ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq971.tif"/> is a family of seminorms on X, then there exists a coarsest locally convex topology T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq972.tif"/> on X such that every element in ℙ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq973.tif"/> is T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq974.tif"/> -continuous, and a local base of closed 0–neighbourhoods is formed by the sets {  x ∈ X  : max 1 ≤ i ≤ n  p i ( x ) ≤ ϵ }   ( ϵ > 0   and   p i ∈ ℙ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq975.tif"/>

Furthermore, T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq976.tif"/> is Hausdorff if and only if for any 0 ≠ x ∈ X there is a p ∈ ℙ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq977.tif"/> such that p(x) ≠ 0.

(2.2) holds for a TVS as shown by the following:

7.c Simple properties of TVS:

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq978.tif"/> be a TVS, let U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq979.tif"/> be a local base and A, B ⊂ X. Then:

(i) A ¯ = ∩ {  A   +   V : V ∈ U X } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq980.tif"/> .

(ii) A + G is open whenever G is open, hence A   +   Int   B   ⊂   Int     ( 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/eq981.tif"/>

115(iii) Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq982.tif"/> be Hausdorff, let K ⊂ X be compact, and let B ⊂ X be closed. If K ∩ B = ϕ then there is an V ∈ U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq983.tif"/> such that ( K  +  V ) ∩ ( B  +  V ) = ϕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq984.tif"/>

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

7.6 Proposition

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq985.tif"/> be a TVS and B ⊂ X.

(a) If B is open then so do ΓB and co B.

(b) Suppose that B is convex (resp. disked), and that Int B ≠ ϕ. Then (7.6.1) λ B ¯ + ( 1 − λ )    Int   B ⊂   Int   B    whenever _    λ ∈ [ 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/eq986.tif"/> (resp. μ B ¯ + ξ Int B ⊂ Int B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq987.tif"/> whenever | μ | + | ξ | ≤ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq988.tif"/> and | μ | ≠ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq989.tif"/> ).

Cosequently, B ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq990.tif"/> and Int B are convex (resp. disked) (7.6.2) B ¯ = Int ¯ B ¯         and _      Int   B = Int   B ¯ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq991.tif"/>

Proof.

(a) An element in ΓB is of the form ∑ i = 1 n   μ i 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/eq992.tif"/> , where x_i ∈ B and ∑ i = 1 n | μ i | = μ ≤ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq993.tif"/> with μ_i ≠ 0. The openness of B ensures that there is a circled 0–neighbourhood V in X such that x i + V ⊂ B   ( 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/eq994.tif"/> . We claim that ∑ i=1 n μ i x i + μ V ⊂ ΓB, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq995.tif"/>

hence the openness of ΓB follows. Indeed, for any y ∈ V, since x i + | μ i | μ i y ∈ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq996.tif"/> , it follows that ∑ i=1 n μ i x i + μ y = ∑ i=1 n μ i ( x i + | μ i | μ i y ) ∈ ΓB, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq997.tif"/>

116which proves our assertion.

The proof of the openness of co B is similar to that of ΓB.

(b) For any fixed λ ∈ [0,1), since λ B ¯ + ( 1 − λ )  Int B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq998.tif"/> is open, it suffices to show that λ B ¯ + ( 1 − λ )    Int   B ⊆ B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq999.tif"/>

Indeed, for any x ∈ Int B, (1 ‒ λ)(Int B −x) is an open 0–neighbourhood, it follows (see (7.c) (i)) that λ B ¯ = λB ¯ ⊆ λB +  ( 1 − λ ) ( Int   B − x ) ⊂ λB + ( 1 − λ ) B − ( 1 − λ ) x ⊆ B − ( 1 − 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/eq1000.tif"/> (on account of the convexity of B), and hence that λ B ¯ + ( 1 − λ )  Int B ⊆ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1001.tif"/> (since x ∈ Int B was arbitrary).

By (7.6.1), it is easily seen that Int B is convex; while the convexity 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/eq1002.tif"/> follows from the continuity of sums and multiplications by scalars.

To prove (7.6.2), we first observe that Int   B ¯ ⊂ B ¯       and     Int   B ⊂ Int   B ¯ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1003.tif"/>

In (7.6.1), letting λ → l,we obtain B ¯ ⊂ Int B ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1004.tif"/> , hence B ¯ = Int B ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1005.tif"/> . On the other hand, to verify that Int  B ¯ ⊂ Int B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1006.tif"/> , it suffices to show that 0 ∈ Int   B ¯ ⇒ 0 ∈ Int   B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1007.tif"/>

117Indeed, there is a circled 0–neighbourhood V such that V ⊆ B ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1008.tif"/> , hence 0 ∈ Int B ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1009.tif"/> (on account of   B ¯ = Int B ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1010.tif"/> and 0 ∈ V ⊂ B ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1011.tif"/> ); consequently, V ∩  Int 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/eq1012.tif"/> . Now we take y ∈ V ∩ Int B, then − y ∈ V ⊂ B ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1013.tif"/> , and apply (7.6.1) to conclude that 0 = 1 2 ( − y ) + 1 2 y ∈ 1 2 B ¯   + 1 2 Int   B ⊂ Int   B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1014.tif"/>

Remark

If B is closed, then ΓB and co B need not be closed. For instance, if B is the closed set in ℝ² which consists of the points (−1,0), (1,0) and the y–axis, then the real absolutely convex hull of B is not closed.

7.7 Definition.

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1015.tif"/> be a TVS and A, B two subsets of X. We say that A absorbs B (or B is absorbed by A) if there exists an λ > 0 such that B ⊂ μ A         ( for    all  | μ | ≥ λ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1016.tif"/>

A subset D of X is said to be bounded if it is absorbed by any 0–neighbourhood in X.

In a normed 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/eq1017.tif"/> , the boundedness of a subset B means that B ⊂ μ 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/eq1018.tif"/> for some μ > 0, hence the definition of boundedness is a generalization of the notion of bounded sets in a normed space; moreover, (5.a) holds for TVS as shown by the following:

7.8 Proposition

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1019.tif"/> be a TVS, and let M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1020.tif"/> be the family of all bounded subsets of X. Then:

(VB1) X = ∪   M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1021.tif"/> .

(VB2)If B ⊂ A and A ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1022.tif"/> then B ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1023.tif"/> .

(VB3) A 1 +   ⋯  + A n ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1024.tif"/> whenever A i ∈ M von ( P )   ( i = 1 , 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/eq1025.tif"/> .

118(VB4) λA ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1026.tif"/> whenever A ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1027.tif"/> and λ ∈ K.

(VB5) The circled hull of any A ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1028.tif"/> belongs to M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1029.tif"/> .

If X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1030.tif"/> is assumed to be a LCS, then (VB5) can be replaced bv (VB5)* ΓA ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1031.tif"/> whenever A ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1032.tif"/> .

The family M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1033.tif"/> is called the von Neumann bornology.

The proof of this result is trivial, hence will be omitted.

Remark

It is clear that A ¯ ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1034.tif"/> whenever A ∈ M von ( P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1035.tif"/> .

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1036.tif"/> , be a TVS. By a fundamental system of bounded sets is meant a family B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1037.tif"/> of P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1038.tif"/> -bounded sets in X such that every bounded set is contained in a suitable member 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/eq1039.tif"/> Clearly, the family of all ( P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1040.tif"/> -closed) circled bounded sets forms a fundamental system of bounded sets. If, in addition, X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1041.tif"/> is a LCS, then the family of all ( P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1042.tif"/> -closed) absolutely convex bounded sets forms a fundamental system of bounded sets.

7.9 Proposition

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1043.tif"/> , be a. TVS and B ⊂ X. Then the following statements are equivalent:

B is bounded.

For any sequence { 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/eq1044.tif"/> in B and any sequence { 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/eq1045.tif"/> in K with λ_n → 0, one has 0 = P − lim n  λ 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/eq1046.tif"/> .

Every sequence in B is bounded.

Proof.

(a) ⇒ (b): Let V be a circled 0–neighbourhood in X. There is an α > 0 such that x_n ∈ αV (for all n ≥ 1), hence λ n x n ∈ λ n α V    ( 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/eq1047.tif"/> . As λ_n → 0, it follows that αλ_n → 0, and hence that there exists a natural number n₀ ≥ 1 such that | α λ 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/eq1048.tif"/> (for all n ≥ n₀); consequently, λ n x n ∈ λ n α V ⊂ V   ( for    al l     n ≥ 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/eq1049.tif"/>

119since V is circled.

(b) ⇒ (c): Suppose that there exists a sequence { 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/eq1050.tif"/> in B which is not bounded. Then there is a circled 0–neighbourhood V such that {  y n : n ≥ 1  }   C   m 2 V      for    all    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/eq1051.tif"/>

For each m ≥ 1, let z m ∈ {  y n : 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/eq1052.tif"/> be such that z_m ∉ m²V. Then { z m } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1053.tif"/> is a sequence in B and 1 m → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1054.tif"/> for which the sequence { 1 m z m } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1055.tif"/> does not converge to 0.

(c) ⇒ (a): Suppose that B is not bounded. Then there exists a circled 0–neighbourhood U such that B     C     n 2 U    ( 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/eq1056.tif"/>

For each n ≥ 1, let x n ∈ B\n 2 U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1057.tif"/> . Then { 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/eq1058.tif"/> is a sequence in B which is not bounded.

In a normed 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/eq1059.tif"/> , the unit ball O_E is bounded and also a convex 0–neighbourhood for the | | ⋅ | | E − top https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1060.tif"/> . The converse is a criterion for a TVS to be normable in the sense that a TVS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1061.tif"/> is normable if its topology P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1062.tif"/> is defined by some norm on X, i.e., if there exists a norm | | ⋅ | | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1063.tif"/> on X such that the balls U r = {  x ∈ X  :     ‖ x ‖ ≤   r  }    ( r > 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1064.tif"/>

form a local base at 0 for P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1065.tif"/> .

7.10 Proposition

A TVS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1066.tif"/> is normable if and only if it is Hausdorff and has a bounded convex P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1067.tif"/> -neighbourhood of 0.

Proof.

The necessity is obvious. To prove the sufficiency, let U be a bounded, convex P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1068.tif"/> -neighbourhood of 0. Then there is a circled 0–neighbourhood V in X such that V ⊂ U, hence W = co V is a bounded, absolutely convex 0–neighbourhood such that 120W ⊂ U. As P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1069.tif"/> is Hausdorff, it follows from the boundedness of W that the gauge p_w of W is a norm on X. [If p_w(x) = 0, then nx ∈ Ker p_w ⊂ W (n ≥ 1), hence 1 n ( nx ) = x → 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1070.tif"/> by the boundedness of W.] Clearly {  n − 1 W : 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/eq1071.tif"/> is a local base for P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1072.tif"/> ; in other words, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1073.tif"/> is defined by the norm p_w.

Remark

The preceding result also tells us that a TVS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1074.tif"/> is normable if and only if it is a LCS and has a bounded 0–neighbourhood.

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1075.tif"/> be a TVS. A net {  x d , d ∈ D  } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1076.tif"/> in X is called a Cauchy net if for any 0–neighbourhood V, there is an d₀ such that x λ − x μ ∈   V     ( for    all     λ , μ ≥ d 0 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1077.tif"/>

A filter base M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1078.tif"/> on X is called a Cauchy filter base if for any 0–neighbourhood V there exists an 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/eq1079.tif"/> such that N − N ⊂ V . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1080.tif"/>

7.11 Definition.

A Hausdorff TVS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1081.tif"/> is said to be:

complete if every Cauchy net in X is convergent;

sequentially complete if every Cauchy sequence in X converges;

quasi–complete if every bounded, Cauchy net in X converges.

7.d The completion:

Let ( X , P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1082.tif"/> be a Hausdorff TVS. There is a complete Hausdorff TVS ( X ˜ , P ˜ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1083.tif"/> containing X as a dense subspace; X ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1084.tif"/> is uniquely up to a topological isomorphism; moreover, if U X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1085.tif"/> is a local base at 0 for P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1086.tif"/> , then {   V ˜ : V ∈ U X   } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1087.tif"/> is a local base for P ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1088.tif"/> , where V ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1089.tif"/> is the P ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1090.tif"/> -closure of V in X ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1091.tif"/> . The space ( X ˜ , P ˜ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1092.tif"/> is called the completion of ( X , P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1093.tif"/> .

7.e <target id="page_121" target-type="page">121</target>Totally bounded sets and compact sets in TVS:

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1094.tif"/> be a TVS and B ⊂ X. B is said to be totally bounded if for any P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1095.tif"/> -neighbourhood V of 0 there is a finite subset { b 1 , ⋯ , b n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1096.tif"/> of B such that B ⊂ ∪ i = 1 n ( b i + V ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1097.tif"/>

A subset B of a Hausdroff TVS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1098.tif"/> is precompact if the closure of B in the completion X ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1099.tif"/> of X is compact. [Clearly, this is equivalent to say that B is regarded as a relatively compact subset of the completion ( X ˜ , P ˜ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1100.tif"/> of ( X , P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1101.tif"/> .] Suppose now that X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1102.tif"/> is a Hausdorff TVS. Then the following assertions are true:

A subset B of X is precompact ⇔ B is totally bounded. ⇔ Every ultrafilter on B is a Cauchy filter.

A subset B of X is compact if and only if it is totally bounded and complete.

The circled hull of a totally bounded (resp.compact) set is totally bounded (resp. compact).

The convex hull (resp.absolutely convex hull)of a finite family of compact convex (resp. compact absolutely convex) sets in X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1103.tif"/> is compact.

7.12 Definition.

A TVS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1104.tif"/> is said to be metrizable if its topology P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1105.tif"/> is metrizable, namely there is a metric d on X such that for any x ∈ X the sets O α ( x ) = {  u ∈ X :    d ( u,x ) < α   }   ( 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/eq1106.tif"/>

form a local base at x for P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1107.tif"/> .

Complete, metrizable LCS are called Fréchet spaces (abbreviated F–spaces).

7.f Metrizable TVS (I):(i)

For a TVS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1108.tif"/> , the following statements are equivalent:

(A) X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1109.tif"/> is metrizable.

122(B) P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1110.tif"/> is Hausdorff and admits a countable local base { V n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1111.tif"/> of circled sets with V n + 1 + V n + 1 ⊂ V n   ( 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/eq1112.tif"/>

(C) P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1113.tif"/> is generated by some F–norm q in the following sense: a mapping q : X → [ 0 , + ∞ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1114.tif"/> satisfies:

(F1) q(x) = 0 if and only if x = 0.

(F2) q(x + y) ≤ q(x) + q(y) (for all x, y ∈ X).

(F3) q(λx) ≤ q(x) (for all x ∈ X and | λ | ≤ 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1115.tif"/> ).

(F4) q(λ_jx) → 0 as λ_j → 0.

(ii) A LCS X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1116.tif"/> is metrizable if and only if P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1117.tif"/> is defined by a sequence { p n : 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/eq1118.tif"/> of seminorms which satisfies the following properties:

(1) p 1 ≤ p 2 ≤ ⋯     ( { p n : n ≥ 1 }   is increasing ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1119.tif"/> .

(2) If p_n(x) = 0 (for all n ≥ 1), then x = 0.

In this case, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1120.tif"/> is determined by an F–norm | ⋅ | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1121.tif"/> , which is given by | x | = ∑ n = 1 ∞   1 2 n p n ( x ) 1 + p n ( x )    ( for    all    x  ∈     X ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1122.tif"/>

The result (2b)(iii), which is a characterization of completeness of normed spaces, can be generalized to the case of metrizable spaces, as shown by the following result.

7.13 Theorem

Let X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1123.tif"/> be a metrizable TVS and let q be an F–norm on X determined by P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1124.tif"/> . Then X P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1125.tif"/> is complete if and only if a formal series ∑ 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/eq1126.tif"/> in X which is absolutely convergent (i.e., ∑ n q ( 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/eq1127.tif"/> ), must be convergent (i.e., u = P − lim n ∑ j = 1 n x j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1128.tif"/> ); in this case, we have q ( u ) ≤ ∑ n q ( 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/eq1129.tif"/>

Proof.

Necessity : The sequence { s n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1130.tif"/> in X, defined by s n = ∑ j = 1 n x j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1131.tif"/> , is a Cauchy 123sequence (since q ( s n + k − s n ) ≤ ∑ i = n + 1 n + k q ( 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/eq1132.tif"/> ), hence converges to a u ∈ X. Clearly the F–norm q is P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1133.tif"/> -continuous,hence q ( u ) = lim n  q ( s n ) ≤ lim n ∑ i = 1 n q ( x i ) = ∑ i q ( 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/eq1134.tif"/>

Sufficiency

Let { y n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1135.tif"/> be a Cauchy sequence in X. Using the Cauchy property, one can construct by induction a subsequence { y n i } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1136.tif"/> of { y n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1137.tif"/> such that q ( y n i + 1 − y n i ) ≤ 1 2 i    ( for    all    i ≥ 1 ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1138.tif"/>

hence ∑ i = 1 ∞ q ( y n i + 1 − y n i ) < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1139.tif"/> . By the assumption, there is a unique z ∈ X such that z = P -   lim r → ∞ ∑ i − 1 r ( y n i + 1 − y n i ) = P -   lim r → ∞ ( y n r + 1 − y 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/eq1140.tif"/> (since ∑ i = 1 r ( y n i + 1 − y n i ) = y n r + 1 − y 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/eq1141.tif"/> ); consequently, P -   lim k  y n k = z + y n 1 = u  ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1142.tif"/>

in other words, the subsequence { y n k } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1143.tif"/> of the Cauchy sequence { y n } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1144.tif"/> is convergent to u, hence u = P − lim n y n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1145.tif"/> .

Banach’s open mapping theorem and closed graph theorem can be generalized to the case of complete metrizable TVS as mentioned in the following:

7.g Complete metrizable TVS:

Let X and Y be TVS. A linear map T : X → Y is said to be almost open if the closure T ( V ) ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1146.tif"/> , in Y, is a 0–neighbourhood in Y whenever V is a 0–neighbourhood in X.

(i) (Banach’s open mapping theorem) Let X and Y be complete metrizable TVS and let T : X → Y be continuous and linear. If T is either almost open or onto, then T is open.

124(ii) (Banach’s closed graph theorm) Let X and Y be complete metrizable TVS and let T : X → Y be linear. If the graph of T is closed, then T is continuous.

(iii) (The uniform boundedness theorem) Let X and Y be F–spaces and R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1147.tif"/> a family of continuous linear maps from X into Y. If R https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1148.tif"/> is bounded pointwisely in the sense that for any x ∈ X, the set { T ( x ) : T ∈ R } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1149.tif"/> is bounded in Y, then for any continuous seminorm q on Y, there exists a continuous seminorm p on X and λ > 0 such that sup T ∈ ℜ  q ( Tx ) ≤ λp ( x )     ( for    all    x  ∈    X ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1150.tif"/>

For any two TVS X and Y, the set consisting of all continuous linear maps (called operators) from X into Y, denoted by L(X,Y), is a vector subspace of L*(X,Y). In particular, if Y = K, then we write X' for L(X,K). X' is called the topological dual of X.

(3.a) and the induced mapping theorem can be generalized to a more general case as shown by the following:

7.h Continuity of linear maps:

Let X and Y be LCS.

(i) The following statements are equivalent for any T ∈ L*(X,Y):

(A) T is continuous at 0.

(B) T is continuous on X.

(C) For any 0–neighbourhood W in Y there is some 0-neighbourhood V in X such that T ( V ) ⊆ W https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1151.tif"/> .

(D) For any continuous seminorm q on Y there exists some continuous seminorm p on X such that q(Tx) ≤ p(x) (for all x ∈ X).

(ii) (Induced mapping theorem) Let Z be another LCS. Given T ∈ L ( X,Y )    and   S ∈ L ( X,Z ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1152.tif"/>

Suppose that S is surjective and that for any continuous seminorm q on Y there is some continuous seminorm r on Z such that q ( Tx ) ≤ r ( Sx )     ( for    all    x  ∈  X ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1153.tif"/>

125Then there exists a unique R ∈ L(Z,Y) such that T = RS.

We conclude this section with a fundamental method for constructing new TVS from given ones.

Let { ( X α , P α ) : α ∈ Λ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1154.tif"/> be a family of TVS. Recall that the product topology Π α P α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1155.tif"/> on Π α X α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1156.tif"/> is the coarsest topology for which all projections π β : Π α X α → X β https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1157.tif"/> are continuous. For each α ∈ Λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1158.tif"/> , let U α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1159.tif"/> be a local base for P α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1160.tif"/> . Then a local base at 0 ∈ Π α X α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1161.tif"/> for Π α P α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1162.tif"/> is given by all sets of the form:

Π α V α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1163.tif"/> with V α ∈ U α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1164.tif"/> and V_α = X_α except for a finite number of indices.

Clearly, each Π α V α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1165.tif"/> is circled and absorbing; it is easy to check that the condition (NS2) of (7.2) holds, hence Π α P α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1166.tif"/> is a vector topology, and thus ( Π α X α , Π α P α ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1167.tif"/> is a TVS. Moreover, it is easily seen that if each P α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1168.tif"/> is locally convex, then so is Π α P α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1169.tif"/> . In particular, if X α = X    and    P α = P     for    all    α ∈ Λ, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1170.tif"/>

then we write 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/eq1171.tif"/> for Π α P α https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1172.tif"/> on the product space X Λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1173.tif"/> .

Let ( X, P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1174.tif"/> be a TVS and M a vector subspace of X. Then the relative topology on M, denoted by P M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1175.tif"/> , is clearly a vector topology. If P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1176.tif"/> is metrizable (or locally convex), then so is P M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1177.tif"/> . Vector subspaces of X, equipped with the relative topologies, are referred to as subspaces of X.

Let Q_M : X → X/M be the quotient map. It is known that the quotient topology on X/M, denoted by P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1178.tif"/> (more precisely P / M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1179.tif"/> ), is the finest topology for which 126Q_M is continuous; in other words, a set B in X/M is P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1180.tif"/> -open if and only if Q M − 1 ( B ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1181.tif"/> is P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1182.tif"/> -open. Moreover, we have the following result:

7.14 Proposition

Let ( X, P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1183.tif"/> be a TVS and M a vector subspace of X.

The quotient topology P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1184.tif"/> on X/M is a vector topology.

The quotient map Q_M is open.

P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1185.tif"/> is Hausdorff if and only if M is closed.

If P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1186.tif"/> is locally convex then so does P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1187.tif"/> .

Proof.

For any P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1188.tif"/> -open set A in X, as Q M -1 ( Q M ( A ) ) = A + M, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1189.tif"/>

it follows that (see (7.c) (ii)) Q_M(A) is P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1190.tif"/> -open, and hence that Q_M is an open map. Accordingly, the family Q M ( U X ) = {  Q M ( V )   :  V ∈ U X } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1191.tif"/>

is a local base at 0 ^ = 0 + M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1192.tif"/> for P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1193.tif"/> . Since Q_M is surjective, the images, under Q_M, of circled, absorbing sets are circled and absorbing. It is easily seen that the condition (NS2) of (7.2) holds, hence P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1194.tif"/> is a vector topology. Therefore (a) and (b) hold.

(c) The necessity is obvious since M = Q M − 1 ( 0 ^ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1195.tif"/> and { 0 ^ } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1196.tif"/> is P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1197.tif"/> -closed. To prove the sufficiency, let 0 ^ ≠ x ^ ∈ 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/eq1198.tif"/> . Then x ^ = Q 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/eq1199.tif"/> for some x ∈ X. Since U = X \ M is an open neighbourhood of x, it follows that Q_M(U) is an P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1200.tif"/> -open set for which 0 ^ ∉ Q M ( U ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1201.tif"/> and x ^ = Q M ( U ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1202.tif"/> . Therefore P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1203.tif"/> is Hausdorff.

(d) Since the image, under Q_M, of any absolutely convex subset of X is absolutely convex, part (d) follows.

7.j Metrizable TVS (II):

Let ( X, P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1204.tif"/> be a complete metrizable TVS and M a 127closed vector subspaces of X. Then the quotient space (X/M, P ∧ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1205.tif"/> ) is a complete metrizable TVS.

Note: In (7.j), the metrizable condition is essential; namely, there is a complete Hausdorff TVS X with a closed vector subspace M such that the quotient space X/M is not complete (see Köthe [1969, ξ31.6]).

7.15 Examples

(a) Let X be a vector space, and U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1206.tif"/> the family of all absolutely convex, absorbing subsets of X. Then U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1207.tif"/> is a filter base on X which satisfies (LC1) and (LC2) of (7.4), hence U https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1208.tif"/> defines a locally convex topology P fin https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1209.tif"/> on X, which is the finest one; thus P fin https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1210.tif"/> is called the finest locally convex topology. It is also clear that P fin https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1211.tif"/> is determined by the family of all seminorms on X.

(b) Let X be a vector space and let X' be a vector subspace of X*. For any f ∈ X', the functional, defined by p f ( x ) = | f ( x ) |     ( for    all    x  ∈  X ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1212.tif"/>

is a seminorm on X. The locally convex topology on X, defined by the family {  p f , f ∈ X ' } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1213.tif"/> of seminorms, is called the weak topology, and denoted by σ ( X,X ' ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1214.tif"/> .

Dually, for any x ∈ X, we define a seminorm q_x on X' by q x ( f ) = | f ( x ) |       ( for    all    f  ∈   X ′ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1215.tif"/>

The locally convex topology on X' determined by { q x : x ∈ X } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1216.tif"/> is called the weak topology (or weak*–topology) on X' and denoted by σ ( X ' ,X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1217.tif"/> .

(c) The space C(Ω) and H(Ω): Let Ω be a non–empty open subset of C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1218.tif"/> (or ℝⁿ). We denote by C(Ω) the vector space of all complex–valued continuous functions on Ω, and by H(Ω) the vector subspace of C(Ω) that consists of holomorphic functions on Ω. It is well–known that there exists a sequence { 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/eq1219.tif"/> of compact sets with Ω = ∪ m = 1 ∞ K m  and    K m   ⊂ Int   K m+1    ( for    all    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/eq1220.tif"/>

128such that each compact subset of Ω is contained in some K_m. For any m ≥ 1, the functional p_m, defined by p m ( x ) = sup   { | x ( t ) |   :  t ∈ K m }       ( for    all   x ∈   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/eq1221.tif"/>

is a seminorm on C(Ω). The metrizable locally convex topology on C(Ω), determined by { p 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/eq1222.tif"/> , is called the compact–open topology. One can show that C(Ω) is complete [since the convergence for the compact–open topology is the convergence uniformly on compact sets K_m, and each (C(K_m),p_m) is a Banach space].

Since sequences of holomorphic functions that converge uniformly on compact sets have holomorphic limits, it follows that H(Ω) is close in C(Ω), and hence that H(Ω) is complete. In complex function theory, a relatively compact subset of C(Ω) is usually called a normal family. Montel’s classical theorem says that any subset of H(Ω) which is uniformly bounded on each compact subset of Ω, is a normal family. It is easily seen from (7.7) that a subset H(Ω), which is uniformly bounded on each compact subset of Ω, is bounded for the compact–open topology. Therefore, Montel’s theorem can be restated by using the language of TVS that any bounded subset of H(Ω) is relatively compact.

(d) (Köthe sequence spaces). A collection P of sequences a = [ a n ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1223.tif"/> of positive numbers is called a Köthe set if it satisfies the following two conditions:

(i) for any a = [a_n], b = [b_n] ∈ P there is an c = [c_n] ∈ P such that a_n,b_n ≥ c_n ( ∀ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1224.tif"/> n ≥ 1);

(ii) for any k ≥ 1, there is some a = [a_n] ∈ P such that a_k > 0.

For any Köthe set P, the set defined by λ ( P ) = { α = [ α n ] ∈ ℂ ℕ : p a ( α ) = ∑ n = 1 ∞ a n | α n | < ∞ , ∀  a = [ a n ] ∈ P } , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1225.tif"/>

is clearly a vector subspace of C N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1226.tif"/> which is called the Köthe sequence space (or the Köthe 129sequence space determined by P). For any a = [ a n ] ∈ P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1227.tif"/> , p a ( α ) = ∑ n = 1 ∞ a n | α n |     for    all     α = [ α n ] ∈    λ ( P ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1228.tif"/>

is clearly a seminorm on λ(P). The family { p n : a ∈ P } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1229.tif"/> determines a Hausdorff locally convex topology on λ(P) which is denoted by T P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1230.tif"/> and called the natural topology determined by P. One can show that ( λ ( P ) , T P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1231.tif"/> is complete.

The set P, defined by P = {   [ n k ] : k = 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/eq1232.tif"/>

is obviously a (countable) Köthe set, the natural topology on λ(P) is determined by a countable family { p k : 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/eq1233.tif"/> of seminorms, where p k ( α ) = ∑ n = 1 ∞ n k | α n |     ( for    all     α = [ α n ] ∈ λ ( P ) ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1234.tif"/> ( λ ( P ) , T P ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq1235.tif"/> is called the Fréchet space of rapidly decreasing sequences, and is denoted by s.

7.k

Let E be a normed space with its Banach dual E', and let { 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/eq1236.tif"/> be a sequence in E. If x_n → 0 for σ ( 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/eq1237.tif"/> and the set { x n : 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/eq1238.tif"/> is relatively compact in E (or sequentially compact),then lim n   | | x 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/eq1239.tif"/> .