<p>We identify the smooth points of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_452_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$L^1(\mu ,X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and provide some necessary and sufficient conditions for left and right symmetry of points with respect to Birkhoff–James orthogonality in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_452_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="153" /> </InlineMediaObject> <EquationSource Format="TEX">$L^p(\mu ,X), 1\le p<\infty$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_452_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu$</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is any complete positive measure and <i>X</i> is a Banach space with some suitable properties.</p>

Some geometric properties of spaces of vector-valued integrable functions

期刊论文
发表日期：

2025年7月25日

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Mohit,
Ranjana Jain

摘要

We identify the smooth points of $L^1(\mu ,X)$ $L^{1} (μ, X)$ , and provide some necessary and sufficient conditions for left and right symmetry of points with respect to Birkhoff–James orthogonality in $L^p(\mu ,X), 1\le p<\infty$ $L^{p} (μ, X), 1 \leq p < \infty$ , where $\mu$ $μ$ is any complete positive measure and X is a Banach space with some suitable properties.