Let E be a directed (i.e., positively generated) ordered vector space endowed with an inner product. In this note, we prove that the following statements are equivalent: <OrderedList> <ListItem> <ItemNumber>(i)</ItemNumber> <ItemContent> E is a vector lattice, and its norm induced by its inner product is a lattice norm. </ItemContent> </ListItem> <ListItem> <ItemNumber>(ii)</ItemNumber> <ItemContent> The metric projection onto the positive cone <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2974_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$E^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>E</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> of E exists, and it is both isotone and subadditive. </ItemContent> </ListItem> </OrderedList> Moreover, in this case, the best approximation to any <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2974_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$x\in E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> from <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2974_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$E^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>E</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation> coincides with its positive part <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2974_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$x^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mo>+</mo> </msup> </math></EquationSource> </InlineEquation>. This result extends previous work on Hilbert lattices to the non-complete setting.

Best Approximation from the Positive Cone of an Inner Product Lattice

期刊论文
发表日期：

2025年10月21日

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Marwen Abdouli,
Karim Boulabiar

摘要

Let E be a directed (i.e., positively generated) ordered vector space endowed with an inner product. In this note, we prove that the following statements are equivalent: (i)

E is a vector lattice, and its norm induced by its inner product is a lattice norm.

(ii)

The metric projection onto the positive cone $E^{+}$ $E^{+}$ of E exists, and it is both isotone and subadditive.

Moreover, in this case, the best approximation to any $x\in E$

x \in E

from $E^{+}$

E^{+}

coincides with its positive part $x^{+}$

x^{+}

. This result extends previous work on Hilbert lattices to the non-complete setting.