Skip to content
Log in
Sponsored Design Multi-Agent AI Systems Using MCP and A2A: Engineer your own Pyt Design Multi-Agent AI Systems Using MCP and A2A: Engineer your own Pyt
Open access

Metrizable Bounded Sets in C(X) Spaces and Distinguished C p (X) Spaces

J. C. Ferrando J. Ka̧kol
2019 · Journal of Convex Analysis · 18 citations · ⚡ 1 influential · 19 references

Abstract

<jats:p> Quite recently W. Ruess [Locally convex spaces not containing <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\ell_{1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , Funct. Approx. Comment. Math. 50 (2014) 351–358] has shown that a wide class of locally convex spaces for which all bounded sets are metrizable enjoy Rosenthal's <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\ell_{1}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -dichotomy. Being motivated by this fact we show that for a Tychonoff space <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> the bounded sets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_{p}(X)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> are metrizable (respectively, the bounded sets of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_{k}(X)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> are weakly metrizable) if and only if <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is countable. If <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is a <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">P</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -space we show that every bounded set in <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_{p}(X)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is metrizable if and only if <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is countable and discrete. The second part of the paper deals with distinguished <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_{p}(X)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> spaces. Among other things we show that <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_{p}(X)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is distinguished if and only if the strong topology of the dual coincides with its strongest locally convex topology, and that <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">C_{p}(X)</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> is always distinguished whenever <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">X</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> </jats:alternatives>

DOI Semantic Scholar
Read PDF

We use cookies to run the site and, with your consent, for analytics and to show ads. See our Cookie Policy.