We study the problem of existence and uniqueness of isometric Banach preduals of a Banach space. We derive necessary and sufficient conditions for the existence of an isometric Banach predual of a Banach space X. Then we focus on the case that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X=\mathcal {F}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a Banach space of scalar-valued functions on a non-empty set <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> and describe those spaces which admit a special isometric Banach predual, namely a strong isometric Banach linearisation, i.e. there are a Banach space Y, a map <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\delta :\Omega \rightarrow Y$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> and an isometric isomorphism <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T:\mathcal {F}(\Omega )\rightarrow Y^{*}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>:</mo> <mi mathvariant="script">F</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msup> <mi>Y</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T(f)\circ \delta = f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>δ</mi> <mo>=</mo> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$f\in \mathcal {F}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Finally, we give necessary and sufficient conditions for Banach spaces <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {F}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">F</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with a strong isometric Banach linearisation to have a (strongly) unique isometric Banach predual.

On linearisation, existence and uniqueness of preduals: the isometric case

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2026年7月14日

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Karsten Kruse

摘要

We study the problem of existence and uniqueness of isometric Banach preduals of a Banach space. We derive necessary and sufficient conditions for the existence of an isometric Banach predual of a Banach space X. Then we focus on the case that $X=\mathcal {F}(\Omega )$ $X = F (Ω)$ is a Banach space of scalar-valued functions on a non-empty set $\Omega $ $Ω$ and describe those spaces which admit a special isometric Banach predual, namely a strong isometric Banach linearisation, i.e. there are a Banach space Y, a map $\delta :\Omega \rightarrow Y$ $δ : Ω \to Y$ and an isometric isomorphism $T:\mathcal {F}(\Omega )\rightarrow Y^{*}$ $T : F (Ω) \to Y^{*}$ such that $T(f)\circ \delta = f$ $T (f) \circ δ = f$ for all $f\in \mathcal {F}(\Omega )$ $f \in F (Ω)$ . Finally, we give necessary and sufficient conditions for Banach spaces $\mathcal {F}(\Omega )$ $F (Ω)$ with a strong isometric Banach linearisation to have a (strongly) unique isometric Banach predual.