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<p>The purpose of this paper is to extend to spaces of nonlinear operators, and also to more general spaces of linear operators, a recent result on lineability of sets of non-injective linear operators. We prove, for quite general spaces <i>A</i> of (linear or nonlinear) maps from an arbitrary set to a sequence space, that, for every <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_458_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(0 \ne f \in A\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≠</mo> <mi>f</mi> <mo>∈</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>, the subset of <i>A</i> of non-injective maps contains an infinite dimensional subspace of <i>A</i> containing <i>f</i>. We provide applications to spaces of linear operators between quasi-Banach spaces, to spaces of linear operators belonging to an operator ideal, and, in the nonlinear setting, to spaces of homogeneous polynomials and to spaces of vector-valued Lipschitz functions on metric spaces.</p>

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Spaceability of sets of non-injective maps

  • 期刊论文

  • 发表日期：

    2025年6月5日

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  • Mikaela Aires,
  • Geraldo Botelho

摘要

The purpose of this paper is to extend to spaces of nonlinear operators, and also to more general spaces of linear operators, a recent result on lineability of sets of non-injective linear operators. We prove, for quite general spaces A of (linear or nonlinear) maps from an arbitrary set to a sequence space, that, for every \(0 \ne f \in A\) 0 ≠ f ∈ A , the subset of A of non-injective maps contains an infinite dimensional subspace of A containing f. We provide applications to spaces of linear operators between quasi-Banach spaces, to spaces of linear operators belonging to an operator ideal, and, in the nonlinear setting, to spaces of homogeneous polynomials and to spaces of vector-valued Lipschitz functions on metric spaces.