Distinguished space

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In functional analysis and related areas of mathematics, distinguished spaces are topological vector spaces (TVSs) having the property that weak-* bounded subsets of their biduals (that is, the strong dual space of their strong dual space) are contained in the weak-* closure of some bounded subset of the bidual.

Suppose that X {\displaystyle X} is a locally convex space and let X ′ {\displaystyle X^{\prime }} and X b ′ {\displaystyle X_{b}^{\prime }} denote the strong dual of X {\displaystyle X} (that is, the continuous dual space of X {\displaystyle X} endowed with the strong dual topology). Let X ′ ′ {\displaystyle X^{\prime \prime }} denote the continuous dual space of X b ′ {\displaystyle X_{b}^{\prime }} and let X b ′ ′ {\displaystyle X_{b}^{\prime \prime }} denote the strong dual of X b ′ . {\displaystyle X_{b}^{\prime }.} Let X σ ′ ′ {\displaystyle X_{\sigma }^{\prime \prime }} denote X ′ ′ {\displaystyle X^{\prime \prime }} endowed with the weak-* topology induced by X ′ , {\displaystyle X^{\prime },} where this topology is denoted by σ ( X ′ ′ , X ′ ) {\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)} (that is, the topology of pointwise convergence on X ′ {\displaystyle X^{\prime }} ). We say that a subset W {\displaystyle W} of X ′ ′ {\displaystyle X^{\prime \prime }} is σ ( X ′ ′ , X ′ ) {\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)} -bounded if it is a bounded subset of X σ ′ ′ {\displaystyle X_{\sigma }^{\prime \prime }} and we call the closure of W {\displaystyle W} in the TVS X σ ′ ′ {\displaystyle X_{\sigma }^{\prime \prime }} the σ ( X ′ ′ , X ′ ) {\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)} -closure of W {\displaystyle W} . If B {\displaystyle B} is a subset of X {\displaystyle X} then the polar of B {\displaystyle B} is B ∘ := { x ′ ∈ X ′ : sup b ∈ B ⟨ b , x ′ ⟩ ≤ 1 } . {\displaystyle B^{\circ }:=\left\{x^{\prime }\in X^{\prime }:\sup _{b\in B}\left\langle b,x^{\prime }\right\rangle \leq 1\right\}.}

A Hausdorff locally convex space X {\displaystyle X} is called a distinguished space if it satisfies any of the following equivalent conditions:

1. If W ⊆ X ′ ′ {\displaystyle W\subseteq X^{\prime \prime }} is a σ ( X ′ ′ , X ′ ) {\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)} -bounded subset of X ′ ′ {\displaystyle X^{\prime \prime }} then there exists a bounded subset B {\displaystyle B} of X b ′ ′ {\displaystyle X_{b}^{\prime \prime }} whose σ ( X ′ ′ , X ′ ) {\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)} -closure contains W {\displaystyle W} .^[1]
2. If W ⊆ X ′ ′ {\displaystyle W\subseteq X^{\prime \prime }} is a σ ( X ′ ′ , X ′ ) {\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right)} -bounded subset of X ′ ′ {\displaystyle X^{\prime \prime }} then there exists a bounded subset B {\displaystyle B} of X {\displaystyle X} such that W {\displaystyle W} is contained in B ∘ ∘ := { x ′ ′ ∈ X ′ ′ : sup x ′ ∈ B ∘ ⟨ x ′ , x ′ ′ ⟩ ≤ 1 } , {\displaystyle B^{\circ \circ }:=\left\{x^{\prime \prime }\in X^{\prime \prime }:\sup _{x^{\prime }\in B^{\circ }}\left\langle x^{\prime },x^{\prime \prime }\right\rangle \leq 1\right\},} which is the polar (relative to the duality ⟨ X ′ , X ′ ′ ⟩ {\displaystyle \left\langle X^{\prime },X^{\prime \prime }\right\rangle } ) of B ∘ . {\displaystyle B^{\circ }.} ^[1]
3. The strong dual of X {\displaystyle X} is a barrelled space.^[1]

If in addition X {\displaystyle X} is a metrizable locally convex topological vector space then this list may be extended to include:

1. (Grothendieck) The strong dual of X {\displaystyle X} is a bornological space.^[1]

All normed spaces and semi-reflexive spaces are distinguished spaces.^[2] LF spaces are distinguished spaces.

The strong dual space X b ′ {\displaystyle X_{b}^{\prime }} of a Fréchet space X {\displaystyle X} is distinguished if and only if X {\displaystyle X} is quasibarrelled.^[3]

Every locally convex distinguished space is an H-space.^[2]

There exist distinguished Banach spaces spaces that are not semi-reflexive.^[1] The strong dual of a distinguished Banach space is not necessarily separable; l 1 {\displaystyle l^{1}} is such a space.^[4] The strong dual space of a distinguished Fréchet space is not necessarily metrizable.^[1] There exists a distinguished semi-reflexive non-reflexive non-quasibarrelled Mackey space X {\displaystyle X} whose strong dual is a non-reflexive Banach space.^[1] There exist H-spaces that are not distinguished spaces.^[1]

Fréchet Montel spaces are distinguished spaces.

• Montel space – Barrelled space where closed and bounded subsets are compact
• Semi-reflexive space

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