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W2150775442 abstract "In this paper, we introduce weakly compact version of the weakly countably determined (<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${mathrm{WCD}}$><mml:mtext>WCD</mml:mtext></mml:math>) property, the strong<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${mathrm{WCD}}$><mml:mtext>WCD</mml:mtext></mml:math>(<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${mathrm{SWCD}}$><mml:mtext>SWCD</mml:mtext></mml:math>) property. A Banach space<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$X$><mml:mi>X</mml:mi></mml:math>is said to be<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${mathrm{SWCD}}$><mml:mtext>SWCD</mml:mtext></mml:math>if there s a sequence (<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A_n $><mml:msub><mml:mi>A</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>) of weak<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$ * $><mml:mo>∗</mml:mo></mml:math>compact subsets of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$X^{ * * } $><mml:msup><mml:mi>X</mml:mi><mml:mrow><mml:mo>∗</mml:mo><mml:mo>∗</mml:mo></mml:mrow></mml:msup></mml:math>such that if<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$K subset X$><mml:mi>K</mml:mi><mml:mo>⊂</mml:mo><mml:mi>X</mml:mi></mml:math>is weakly compact, there is an<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$left( {n_m } right) subset N$$><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>⊂</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math>such that<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$K subset bigcapnolimits_{m = 1}^infty {A_{n_m } subset X} $><mml:mi>K</mml:mi><mml:mo>⊂</mml:mo><mml:msubsup><mml:mo>⋂</mml:mo><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>∞</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:msub><mml:mo>⊂</mml:mo><mml:mi>X</mml:mi></mml:mrow></mml:math>. In this case, (<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$A_n $><mml:msub><mml:mi>A</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math>) is called a strongly determining sequence for<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$X$><mml:mi>X</mml:mi></mml:math>. We show that<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${mathrm{SWCG}} Rightarrow {mathrm{SWCD}}$><mml:mtext>SWCG</mml:mtext><mml:mo>⇒</mml:mo><mml:mtext>SWCD</mml:mtext></mml:math>and that the converse does not hold in general. In fact,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$X$><mml:mi>X</mml:mi></mml:math>is a separable<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=${mathrm{SWCD}}$><mml:mtext>SWCD</mml:mtext></mml:math>space if and only if (<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$X$><mml:mi>X</mml:mi></mml:math>, weak) is an<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$aleph _0 $ ><mml:msub><mml:mi>ℵ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>-space. Using<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$c_0 $><mml:msub><mml:mi>c</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>for an example, we show how weakly compact structure theorems may be used to construct strongly determining sequences." @default.
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W2150775442 date "1995-01-01" @default.
W2150775442 modified "2023-10-14" @default.
W2150775442 title "The strongWCDproperty for Banach spaces" @default.
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