For locally convex spaces, we systematize several known equivalent definitions of Fréchet (Gâteaux) Differentiability Spaces and Asplund (Weak Asplund) Spaces. As an application, we extend the classical Mazur’s theorem as follows: Let E be a separable Baire locally convex space and let Y be the product <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\prod _{\alpha \in A} E_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∏</mo> <mrow> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> <msub> <mi>E</mi> <mi>α</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> of any family of separable Fréchet spaces; then the product <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$E \times Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mo>×</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> is Weak Asplund. Also, we prove that the product Y of any family of Banach spaces <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(E_{\alpha })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>α</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is an Asplund locally convex space if and only if each <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$E_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> is Asplund. Analogues of both results are valid under the same assumptions, if Y is the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation>-product of any family <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(E_{\alpha })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>α</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

On the product of Weak Asplund locally convex spaces

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2025年6月9日

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Jerzy Kakol,
Arkady Leiderman

摘要

For locally convex spaces, we systematize several known equivalent definitions of Fréchet (Gâteaux) Differentiability Spaces and Asplund (Weak Asplund) Spaces. As an application, we extend the classical Mazur’s theorem as follows: Let E be a separable Baire locally convex space and let Y be the product $\prod _{\alpha \in A} E_{\alpha }$ $\prod_{α \in A} E_{α}$ of any family of separable Fréchet spaces; then the product $E \times Y$ $E \times Y$ is Weak Asplund. Also, we prove that the product Y of any family of Banach spaces $(E_{\alpha })$ $(E_{α})$ is an Asplund locally convex space if and only if each $E_{\alpha }$ $E_{α}$ is Asplund. Analogues of both results are valid under the same assumptions, if Y is the $\Sigma $ $Σ$ -product of any family $(E_{\alpha })$ $(E_{α})$ .