Dual Pairs and the Weak Topology | 16 | Introductory Theory of Topolog

ABSTRACT

We begin with the following: 16.1 Definition.

Let X and Y be vector spaces over K, and suppose that there exists a bilinear functional ψ on X × Y, satisfying the following separation conditions:

(S1) if ψ(x₀, y) = 0 (for all y ∈ Y) then x₀ = 0;

(S2) if ψ(x, y₀) = 0 (for all x ∈ X) then y₀ = 0.

Then the triple (X, Y, ψ) is called a dual pair (or a duality), and ψ is called the canonical bilinear functional of the duality. Usually, we write < x,y > = υ ( x,y ) ( for all x ∈ X and y ∈ Y ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2599.tif"/>

while the triple (X, Y, ψ) is more conveniently denoted by < X,Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2600.tif"/> .

Remark

Because of the separation condition, the map y → < ⋅ , 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/eq2601.tif"/>

is linear and injective; symmetrically the map x → < x , ⋅ > : X → Y * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2602.tif"/>

is linear and injective. Thus we identify Y with a vector subspace of X* which separates points on X, and symmetrically X with a vector subspace of Y* which separates points on Y.

16.2 Examples

(a) Let X be a vector space and X* the algebraic dual of X. Then < 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/eq2603.tif"/> is a dual pair under the natural bilinear form (16.2.1) < u, x * > = x * ( u ) ( for all u ∈ X and 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/eq2604.tif"/>

206Moreover, if Y is a vector subspace of X* which separates points on X, then the restriction on X × Y of the natural bilinear form (16.2.1) induces a duality between X and Y. In particular, 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/eq2605.tif"/> ) is a LCS with the topological dual X', then < 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/eq2606.tif"/> is a dual pair under the natural bilinear form: < u, x ' > = x ' ( u ) ( for all u ∈ X and 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/eq2607.tif"/>

(b) Bornological duality. Let (X, G https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2608.tif"/> ) be a CBVS with the bornological dual X^x. If X^x separates points on X, then < 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/eq2609.tif"/> is a duality under the natural bilinear form < u, f > = f ( u ) ( for all u ∈ X and f ∈ 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/eq2610.tif"/>

in this case, < 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/eq2611.tif"/> is referred to as the bornological duality.

16.3 Definition.

For a given dual pair < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2612.tif"/> , the weak topology on X, denoted by σ(X,Y), is defined by the family { p y : y ∈ Y } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2613.tif"/> of seminorms on X, where p y ( x ) = | < x,y > | ( 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/eq2614.tif"/>

Symmetrically we define the weak topology on Y, denoted by σ(Y,X). Subsets of X which are closed (resp. open, compact, totally bounded etc.) for σ(X,Y) are called weakly closed (resp. weakly open, weakly compact, weakly totally bounded, etc.)

By (S1), σ(X,Y) is a Hausdorff locally convex topology on X which is the coarsest topology such that all linear functionals { < ⋅ , y > : y ∈ Y } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2615.tif"/> are continuous, hence Y can be identified with a vector subspace of ( X , σ ( X, Y ) ) ' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2616.tif"/> ; they are equal as shown by the following:

16.4 Lemma

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2617.tif"/> be a dual pair. Then Y ≃ ( X , σ ( X , Y ) ) ' ( algebraically isomorphic ) ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2618.tif"/>

207 namely, an f ∈ X* is of σ(X,Y)–continuous if and only if there exists a unique y ∈ Y such that (1) f ( x ) = < x, y > ( 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/eq2619.tif"/>

(Hence we always identify Y with ( X , σ ( X, Y ) ) ' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2620.tif"/> , and say that Y is the topological dual of ( X , σ ( X, Y ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2621.tif"/> .)

Proof.

Let f ∈ X* be σ(X, Y)–continuous. Then there exists a λ > 0 and { y 1 , ⋯ , y n } ⊂ Y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2622.tif"/> such that | f ( x ) | ≤ λ max 1 ≤ i ≤ n | < x,y i > | ( 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/eq2623.tif"/>

so that (2) < x,y i > = 0 ( i = 1 , ⋯ , n ) ⇒ f ( 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/eq2624.tif"/>

It then follows from a well–known result that there are μ 1 , ⋯ , μ 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/eq2625.tif"/> such that f ( x ) = ∑ i − 1 n μ i < x, y i > = < x , ∑ i = 1 n μ i y i > ( 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/eq2626.tif"/>

thus the result follows.

16.5 Corollary

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/eq2627.tif"/> ) be a LCS with the topological dual X'. Then X ' = ( X , 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/eq2628.tif"/> .

16.a

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2629.tif"/> be a dual pair and Y₁ a vector subspace of Y. Then Y₁ is σ(Y, X)–dense in Y if and only if Y₁ separates points on X in the sense that < x 0 , y > = 0 ( for all y ∈ Y 1 ) ⇒ x 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/eq2630.tif"/>

In particular, 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/eq2631.tif"/> ) is a LCS with the topological dual X', then X' is 208 σ(X*, X)–dense in X*.

16.6 Proposition

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2632.tif"/> be a duality. A subset B of X is σ(X, Y)–bounded if and only if B is σ(X, Y)–totally bounded.

Proof.

The sufficiency is trivial. To prove the necessity, let V be a σ(X, Y)–neighbourhood of 0, and let y ∈ Y be such that W = { x ∈ X : | < x, y > | ≤ 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/eq2633.tif"/>

Since B is σ(X, Y)–bounded, the set { < b, y > : 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/eq2634.tif"/> is bounded in K, hence relatively compact in K; consequently, there exists 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/eq2635.tif"/> of B such that { < b, y > : b ∈ B } ⊂ ∪ i = 1 n ( < b i , y > + U K ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2636.tif"/>

It then follows that B ⊂ ∪ i = 1 n ( b i + W ) ⊂ ∪ 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/eq2637.tif"/>

and hence that B is σ(X, Y)–totally bounded.

As a consequence of (16.6) and (2.11), we have the following:

16.b

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/eq2638.tif"/> is finite–diminsional if and only if | | ⋅ | | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2639.tif"/> –top coincides with σ(E, E').

16.7 Definition.

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2640.tif"/> be a dual pair and ϕ ≠ B ⊂ X. The set, defined by B ∘ = { y ∈ Y : Re <b,y> ≤ 1 ( for all 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/eq2641.tif"/>

is called the polar of B (taken in Y or w.r.t. < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2642.tif"/> ); while the set, defined by B a = { y ∈ Y : | < b, y > | ≤ 1 ( for all 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/eq2643.tif"/>

209is called the absolute polar of B (taken in Y or w.r.t. < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2644.tif"/> ). (When there is need to specify the polar taken in Y, we shall write B°(Y) for B°.) The bipolar of B, which is a subset of X, is defined to be the polar of B° (taken in X), and denoted by B⁰⁰.

16.8 Lemma

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2645.tif"/> be a dual pair and ϕ ≠ B ⊂ X.

B° (resp. B^a) is a σ(Y.X)–closed convex (resp. σ(Y,X)–closed, disked) subset of Y containing 0.

(λB)° = λ^-1B° (for all λ ≠ 0).

If A ⊂ B ⊂ X then B° ⊂ A°.

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/eq2646.tif"/> and 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/eq2647.tif"/> .

If B_α ⊂ X (α ∈ A), then ( ∪ α ∈ Λ B α ) 0 = ∩ α B α 0 and _ ( ∩ B α ) 0 ⊃ co ¯ ( ∪ α B 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/eq2648.tif"/>

where the bar denotes the σ(Y, X)–closure.

The proof is straighforward, hence will be omitted.

16.9 Theorem (Bipolar theorem)

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2649.tif"/> be a dual pair and B ⊂ X. Then B ° ° = CO ¯ ( B ∪ { 0 } ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2650.tif"/> (where the bar denotes the σ(X,Y)–closure).

Proof.

Let B 1 = CO ¯ ( B ∪ { 0 } ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2651.tif"/> . Then B₁ ⊂ B ° ° https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2652.tif"/> . If B₁ ≠ B ° ° https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2653.tif"/> , then we take x 0 ∈ B ° ° \ B 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2654.tif"/> As Y = ( X , σ ( X , Y ) ) ' https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2655.tif"/> , it follows from the strong separation theorem that there is an 0 ≠ y ∈ Y such that sup { Re < x,y > : x ∈ B 1 } < 1 < Re < x 0 ,y > ; https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2656.tif"/>

210in particular, (1) sup { Re < b,y > : b ∈ B } < 1 < Re < x 0 ,y > , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2657.tif"/>

The left side of (1) shows that y ∈ B°, while the right side of (1), together with x₀ ∈ B ° ° https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2658.tif"/> shows that 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/eq2659.tif"/> = B° (by (16.8) (d)). Thus we arrive at a contradiction.

16.10 Corollary

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2660.tif"/> be a dual pair and B ⊂ X. Then B ° = ( co ¯ ( B ∪ { 0 } ) ) ° = ( co ( B ∪ { 0 } ) ) ° . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2661.tif"/>

Proof.

As 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/eq2662.tif"/> , the result follows immediately from (16.9).

16.c

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2663.tif"/> be a dual pair and let { B i : 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/eq2664.tif"/> be a family of convex subsets of X with 0 ∈ ∩ i B i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2665.tif"/> . Then ( ∩ i B i ) ° = CO ¯ ( ∩ i B i ° ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2666.tif"/> if and only if ∩ i B ¯ i = ∩ i B i ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2667.tif"/> .

16.11 Proposition

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2668.tif"/> be a dual pair and B ⊂ X.

(a) B is σ(X, Y)–bounded if and only if B° is an absorbing subset of Y.

(b) B° is σ(Y.X)–bounded if and only if B ° ° https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2669.tif"/> is an absorbing subset of X.

(c) Suppose that 0 ∈ B = co B ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2670.tif"/> . Then B is σ(X.Y)–complete if and only if every f ∈ Y* which is bounded on B° is represented by some element in X.

Proof.

(a) Follows from the following equivalent statements:

B is σ(X,Y)–bounded

⇔ for any y ∈ Y, sup { | < b , y > | : b ∈ B } = λ y < ∞ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2671.tif"/>

⇔ y ∈ λ_yB°

⇔B° is absorbing in Y.

(b) Follows from part (a) in view of the bipolar theorem.

(c) We first notice that (Y*,σ(Y*, Y)) is complete and that σ ( X , Y ) = σ ( 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/eq2672.tif"/> , here we identify X as a vector subspace of Y*. [In fact, Y* is closed in the product 211space K^Y, while the product space is K^Y complete for the product topology]. Thus we have the following equivalent statements:

B is σ(X,Y)–complete

⇔ B is σ(Y, Y)–closed

⇔ B = (B°)°(Y*) (by the bipolar theorem and 0 ∈ B = coB ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2673.tif"/> )

⇔ every g ∈ Y* which is bounded on B° is represented by some element of X.

16.12 Theorem (Smulia)

Let < X, Y> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2674.tif"/> be a dual pair, and let B ⊂ X be such that 0 ∈ B = co ¯ B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2675.tif"/> (the σ(X,Y)–closure). Then B is σ(X.Y)–compact if and only if B° is absorbing in Y and every g ∈ Y* which is bounded on B° is represented bv some element in X.

Proof.

Follows from the following equivalent statements:

B is σ(X,Y)–compact

⇔ B is σ(X,Y)–totally bounded and σ(X,Y)–complete

⇔ B is σ(X,Y)–bounded and σ(X,Y)–complete

⇔ B° is absorbing in Y and every g ∈ Y* which is bounded on B° is represented by some element in X.

16.13 Theorem. (Alaoglu–Bourbaki)

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/eq2676.tif"/> ) be a LCS. Then the polar of any 0–neighbourhood V in X is σ(X'.X)–compact.

Proof.

Without loss of generality one can assume that V = co ¯ V https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2677.tif"/> , so 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/eq2678.tif"/> (by (16.9)). Any f ∈ X* which is bounded on V must be continuous, hence it can be represented by some element of X', it follows from (16.11)(c) that V° is σ(X',X)–complete. On the other hand, V is absorbing in X, hence V° is σ(X',X)–bounded (by (16.6)(b)), thus V° is σ(X',X)–totally bounded (by (16.6)).Consequently, V° is σ(X',X)–compact.

212Let (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/eq2679.tif"/> ) be a LCS. Recall that a subset D of X' is equicontinuous if there exists 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/eq2680.tif"/> _X such that V ⊂ ∩ f ∈ D { x ∈ X : | f ( 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/eq2681.tif"/>

or, equivalently, D ⊂ V°. Therefore, Alaoglu–Bourbaki’s theorem asserts that 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/eq2682.tif"/> –equicontinuous subset of X' is relatively σ(X',X)–compact (and a fortiori σ(X',X)–bounded).

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/eq2683.tif"/> is called a Banach dual space if there exists a normed space (F, q) such that ( E, | | ⋅ | | ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2684.tif"/> and the Banach dual (F',q') of (F, q) are isometric, that is, E = F' and | | ⋅ | | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2685.tif"/> is the dual norm q' of q.

Every Banach dual space must be complete, but there are B–spaces which are not Banach dual spaces. On the other hand, if ( E, | | ⋅ | | ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2686.tif"/> is a Banach dual space, then Alaoglu–Bourbaki’s theorem ensures that there is a Hausdorff 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/eq2687.tif"/> on E (that is, the topology σ(E,F)) such that the closed unit ball U E = { x ∈ E : | | 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/eq2688.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/eq2689.tif"/> –compact. The converse is true as shown by the following result.

16.14 Theorem (Ng)

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/eq2690.tif"/> is a Banach dual space if and only if there exists a Ilausdorff 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/eq2691.tif"/> on E such that the closed unit ball U_E in E 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/eq2692.tif"/> –compact.

Proof.

It has only to verify the sufficiency. Consider the following vector subspace of E*: F = { f ∈ E * : f is P − continuous on 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/eq2693.tif"/>

213Then ( E , P ) ' ⊆ F ⊆ 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/eq2694.tif"/> (the inclusion F ⊆ E' follows from 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/eq2695.tif"/> –compactness of U_E). F is closed in the Banach dual E' of ( E, | | ⋅ | | ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2696.tif"/> , hence (F, q) is a Banach space, where q the restriction on F of the dual norm of | | ⋅ | | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2697.tif"/> , i.e., q ( f ) = ‖ f ‖ = sup { | f ( x ) | : x ∈ U E } ( for all f ∈ F ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2698.tif"/>

Let (F',q*) be the Banach dual of (F,q), where q* is the dual norm of q. It remains to show that ( E, | | ⋅ | | ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2699.tif"/> and (F',q*) are isometric.

In fact, the map J_F : E → F', defined by ( J F x ) f = f ( x ) ( for all f ∈ F ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2700.tif"/>

is linear, injective (since F separates points of E) and q * ( J F x ) = sup { | ( J F x ) f | : f ∈ F, q ( f ) ≤ 1 } ≤ ‖ x ‖ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2701.tif"/>

Thus it has only to verify that J F ( U E ) = U F ' = { υ : F ' : q * ( υ ) ≤ 1 } . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2702.tif"/>

To this end, we first notice that J_F(U_E) is σ(F',F)–compact (the restriction of J_F on U_E is continuous 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/eq2703.tif"/> and σ(F',F) and the definition of F); hence 214the bipolar theorem ensures that J F ( U E ) = ( J F ( 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/eq2704.tif"/>

where the polar is taken with respect to the duality < F , F ' > https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2705.tif"/> . Observe that ( J F ( U E ) ) ∘ = { f ∈ F : | f ( x ) | ≤ 1 ( for all x ∈ U E ) } = F ∩ { x ' ∈ E ' : ‖ x ' ‖ ≤ 1 } = { f ∈ F : q ( f ) ≤ 1 } = the closed unit ball U F 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/eq2706.tif"/>

Therefore J_F(U_E) = U F ∘ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq2707.tif"/> = U_F,, which obtains our assertion.

The proof of the preceding result is taken from Ng [1971 b], but the main idea of the proof is essentially due to Edwards [1964]. While the usual form of Dixmier’s theorem states as follows : A Banach space E is a Banach dual space if and only if there exists a vector subspace F of E', which separates points of E, such that U_E is σ(E,F)–compact.