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• W1560175257 abstract "We construct a weakly null normalized sequence <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace f Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding=application/x-tex>{f_i}_{i=1}^{infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L 1> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=application/x-tex>L_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that for each <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>varepsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the Haar basis is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis 1 plus epsilon right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(1+varepsilon )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivalent to a block basis of every subsequence of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace f Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding=application/x-tex>{f_i}_{i=1}^{infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, the sequence <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-brace f Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> <mml:annotation encoding=application/x-tex>{f_i}_{i=1}^{infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no unconditionally basic subsequence. This answers a question raised by Bernard Maurey and H. P. Rosenthal in 1977. A similar example is given in an appropriate class of rearrangement invariant function spaces." @default.
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• W1560175257 date "2006-09-19" @default.
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• W1560175257 title "Weakly null sequences in 𝐿₁" @default.
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• W1560175257 doi "https://doi.org/10.1090/s0894-0347-06-00548-0" @default.
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