We discuss some properties of the Banach-valued sequence space <mml:math alttext="$\ell_p[X]$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x2113;</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mi>X</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math><mml:math alttext="$(1\leq p &#x3C; \infty)$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2264;</mml:mo><mml:mi>p</mml:mi><mml:mo>&#x3C;</mml:mo><mml:mi>&#x221E;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>, the space of weakly <mml:math alttext="$p$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-summable sequences on a Banach space <mml:math alttext="$X$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>. For example, we characterize the reflexivity of <mml:math alttext="$\ell_{p}[X]$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x2113;</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mi>X</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math>, convergent sequences on <mml:math alttext="$\ell_{p}[X]$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x2113;</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mi>X</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math>, and compact subsets of <mml:math alttext="$\ell_{p}[X]$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x2113;</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mi>X</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math>.

Some properties of Banach-valued sequence spaces <mml:math alttext="$\ell_{p}[X]$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℓ</mml:mi><mml:mi>p</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mi>X</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:math>

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International Journal of Mathematics and Mathematical Sciences

Some properties of Banach-valued sequence spaces ℓp[X]

Abstract

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