tempRecords.jsonl

{"src": "06696052_tex.tex", "susp": "1309.2020_TEX.tex"}
{"parentHTMLfraction": ["<p class=\"ltx_p\"><div style=\"background-color: rgb(188, 203, 235);\">We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m1\" class=\"ltx_Math\" alttext=\"{}^{*}\" display=\"inline\"><msup><mi></mi><mo>*</mo></msup></math> null sequence of functionals converges uniformly to zero</div>. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m2\" class=\"ltx_Math\" alttext=\"L\\,\" display=\"inline\"><mpadded width=\"+1.7pt\"><mi>L</mi></mpadded></math>-weakly compact sets coincide. In particular, in terms of almost Dunford-Pettis operators into <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m3\" class=\"ltx_Math\" alttext=\"c_{0}\" display=\"inline\"><msub><mi>c</mi><mn>0</mn></msub></math>, we give an operator characterization of those <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m4\" class=\"ltx_Math\" alttext=\"\\sigma\" display=\"inline\"><mi>σ</mi></math>- Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m5\" class=\"ltx_Math\" alttext=\"\\sigma\" display=\"inline\"><mi>σ</mi></math>-Dedekind Banach lattice <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m6\" class=\"ltx_Math\" alttext=\"E\" display=\"inline\"><mi>E</mi></math>, every relatively weakly compact set in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m7\" class=\"ltx_Math\" alttext=\"E\" display=\"inline\"><mi>E</mi></math> is almost limited if and only if every continuous linear operator <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m8\" class=\"ltx_Math\" alttext=\"T:E\\rightarrow c_{\\,0}\" display=\"inline\"><mrow><mi>T</mi><mo>:</mo><mrow><mi>E</mi><mo>→</mo><msub><mi>c</mi><mn> 0</mn></msub></mrow></mrow></math> is an almost Dunford-Pettis operator.</p>"], "htmlFraction": ["We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m1\" class=\"ltx_Math\" alttext=\"{}^{*}\" display=\"inline\"><msup><mi></mi><mo>*</mo></msup></math> null sequence of functionals converges uniformly to zero"], "text": ["We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak* null sequence of functionals converges uniformly to zero"], "count": ["1"], "color": ["#BCCBEB"], "csrfmiddlewaretoken": ["{{ csrf_token }}"]}
{"parentHTMLfraction": ["<p class=\"ltx_p\"><div style=\"background-color: rgb(188, 203, 235);\">We introduce and study the class of weak almost limited operators.</div> We\nestablish a characterization of pairs of Banach lattices <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m1\" class=\"ltx_Math\" alttext=\"E\" display=\"inline\"><mi>E</mi></math>, <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m2\" class=\"ltx_Math\" alttext=\"F\" display=\"inline\"><mi>F</mi></math> for which\nevery positive weak almost limited operator <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"m3\" class=\"ltx_Math\" alttext=\"T:E\\rightarrow F\" display=\"inline\"><mrow><mi>T</mi><mo>:</mo><mrow><mi>E</mi><mo>→</mo><mi>F</mi></mrow></mrow></math> is almost\nlimited (resp. almost Dunford-Pettis). As consequences, we will give some\ninteresting results.</p>"], "htmlFraction": ["We introduce and study the class of weak almost limited operators."], "text": ["We introduce and study the class of weak almost limited operators."], "count": ["2"], "color": ["#BCCBEB"], "csrfmiddlewaretoken": ["{{ csrf_token }}"]}
{"parentHTMLfraction": ["<p class=\"ltx_p\"><div style=\"background-color: rgb(237, 189, 188);\">Let us recall that a bounded subset <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m1\" class=\"ltx_Math\" alttext=\"A\" display=\"inline\"><mi>A</mi></math> of <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m2\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> is called a <span class=\"ltx_text ltx_font_italic\">Dunford-Pettis set</span> (resp. a <span class=\"ltx_text ltx_font_italic\">limited set</span>) in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m3\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> if each weakly null sequence in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m4\" class=\"ltx_Math\" alttext=\"X^{*}\" display=\"inline\"><msup><mi>X</mi><mo>*</mo></msup></math> (resp. weak<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m5\" class=\"ltx_Math\" alttext=\"{}^{*}\" display=\"inline\"><msup><mi></mi><mo>*</mo></msup></math> null sequence in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m6\" class=\"ltx_Math\" alttext=\"X^{*}\" display=\"inline\"><msup><mi>X</mi><mo>*</mo></msup></math>) converges uniformly to zero on <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m7\" class=\"ltx_Math\" alttext=\"A\" display=\"inline\"><mi>A</mi></math>. Clearly, every limited set in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m8\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> is a Dunford-Pettis set, but the converse is not true in general</div>. We say that <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m9\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> has the <span class=\"ltx_text ltx_font_italic\">Dunford-Pettis property </span>whenever <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m10\" class=\"ltx_Math\" alttext=\"x_{n}\\xrightarrow{w}0\" display=\"inline\"><mrow><msub><mi>x</mi><mi>n</mi></msub><mover accent=\"true\"><mo>→</mo><mo>𝑤</mo></mover><mn>0</mn></mrow></math> in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m11\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> and <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m12\" class=\"ltx_Math\" alttext=\"f_{n}\\xrightarrow{w}0\" display=\"inline\"><mrow><msub><mi>f</mi><mi>n</mi></msub><mover accent=\"true\"><mo>→</mo><mo>𝑤</mo></mover><mn>0</mn></mrow></math> in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m13\" class=\"ltx_Math\" alttext=\"X^{*}\" display=\"inline\"><msup><mi>X</mi><mo>*</mo></msup></math> imply <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m14\" class=\"ltx_Math\" alttext=\"\\lim_{\\,n}f_{n}(x_{n})=0\" display=\"inline\"><mrow><mrow><msub><mo>lim</mo><mpadded lspace=\"1.7pt\" width=\"+1.7pt\"><mi>n</mi></mpadded></msub><mo>⁡</mo><mrow><msub><mi>f</mi><mi>n</mi></msub><mo>⁢</mo><mrow><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy=\"false\">)</mo></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>, equivalently, every relatively weakly compact set in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m15\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> is a Dunford-Pettis set, alternatively, every weakly compact operator <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m16\" class=\"ltx_Math\" alttext=\"T:X\\rightarrow c_{0}\" display=\"inline\"><mrow><mi>T</mi><mo>:</mo><mrow><mi>X</mi><mo>→</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></mrow></math> is a Dunford-Pettis operator. If all limited sets in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m17\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> are relatively compact, then <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m18\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> is said to be a <span class=\"ltx_text ltx_font_italic\">Gelfand-Phillips space</span>. It is well-known that all separable Banach spaces and all weakly compactly generated spaces are Gelfand-Phillips spaces. Note that a <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m19\" class=\"ltx_Math\" alttext=\"\\sigma\" display=\"inline\"><mi>σ</mi></math>-Dedekind complete Banach lattice <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m20\" class=\"ltx_Math\" alttext=\"E\" display=\"inline\"><mi>E</mi></math> is a Gelfand-Phillips space if and only if the norm of <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m21\" class=\"ltx_Math\" alttext=\"E\" display=\"inline\"><mi>E</mi></math> is order continuous (cf. <cite class=\"ltx_cite ltx_citemacro_cite\">[<a href=\"#bib.bib7\" title=\"\" class=\"ltx_ref\">7</a>]</cite>). <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m22\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> has the <span class=\"ltx_text ltx_font_italic\">Dunford-Pettis<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m23\" class=\"ltx_Math\" alttext=\"{}^{*}\" display=\"inline\"><msup><mi></mi><mo mathvariant=\"normal\">*</mo></msup></math> property</span> (the DP<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m24\" class=\"ltx_Math\" alttext=\"{}^{*}\" display=\"inline\"><msup><mi></mi><mo>*</mo></msup></math> property for short ) whenever every relatively weakly compact set in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m25\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> is limited, in other words, for any weakly null sequence <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m26\" class=\"ltx_Math\" alttext=\"(x_{n})\" display=\"inline\"><mrow><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy=\"false\">)</mo></mrow></math> in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m27\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> and any weak<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m28\" class=\"ltx_Math\" alttext=\"{}^{*}\" display=\"inline\"><msup><mi></mi><mo>*</mo></msup></math> null sequence <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m29\" class=\"ltx_Math\" alttext=\"(f_{n})\" display=\"inline\"><mrow><mo stretchy=\"false\">(</mo><msub><mi>f</mi><mi>n</mi></msub><mo stretchy=\"false\">)</mo></mrow></math> in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m30\" class=\"ltx_Math\" alttext=\"X^{*}\" display=\"inline\"><msup><mi>X</mi><mo>*</mo></msup></math>, <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m31\" class=\"ltx_Math\" alttext=\"\\lim_{\\,n}f_{n}(x_{n})=0\" display=\"inline\"><mrow><mrow><msub><mo>lim</mo><mpadded lspace=\"1.7pt\" width=\"+1.7pt\"><mi>n</mi></mpadded></msub><mo>⁡</mo><mrow><msub><mi>f</mi><mi>n</mi></msub><mo>⁢</mo><mrow><mo stretchy=\"false\">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy=\"false\">)</mo></mrow></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math>. The DP<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m32\" class=\"ltx_Math\" alttext=\"{}^{*}\" display=\"inline\"><msup><mi></mi><mo>*</mo></msup></math> property, introduced first by Borwein, Fabian and Vanderwerff <cite class=\"ltx_cite ltx_citemacro_cite\">[<a href=\"#bib.bib4\" title=\"\" class=\"ltx_ref\">4</a>]</cite>, is stronger than the Dunford-Pettis property. Carrión, Galindo and Lourenço <cite class=\"ltx_cite ltx_citemacro_cite\">[<a href=\"#bib.bib9\" title=\"\" class=\"ltx_ref\">9</a>]</cite> showed that <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m33\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> has the DP<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m34\" class=\"ltx_Math\" alttext=\"{}^{*}\" display=\"inline\"><msup><mi></mi><mo>*</mo></msup></math> property if, and only if, every continuous linear operator <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m35\" class=\"ltx_Math\" alttext=\"T:X\\rightarrow c_{0}\" display=\"inline\"><mrow><mi>T</mi><mo>:</mo><mrow><mi>X</mi><mo>→</mo><msub><mi>c</mi><mn>0</mn></msub></mrow></mrow></math> is a Dunford-Pettis operator.</p>"], "htmlFraction": ["Let us recall that a bounded subset <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m1\" class=\"ltx_Math\" alttext=\"A\" display=\"inline\"><mi>A</mi></math> of <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m2\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> is called a <span class=\"ltx_text ltx_font_italic\">Dunford-Pettis set</span> (resp. a <span class=\"ltx_text ltx_font_italic\">limited set</span>) in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m3\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> if each weakly null sequence in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m4\" class=\"ltx_Math\" alttext=\"X^{*}\" display=\"inline\"><msup><mi>X</mi><mo>*</mo></msup></math> (resp. weak<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m5\" class=\"ltx_Math\" alttext=\"{}^{*}\" display=\"inline\"><msup><mi></mi><mo>*</mo></msup></math> null sequence in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m6\" class=\"ltx_Math\" alttext=\"X^{*}\" display=\"inline\"><msup><mi>X</mi><mo>*</mo></msup></math>) converges uniformly to zero on <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m7\" class=\"ltx_Math\" alttext=\"A\" display=\"inline\"><mi>A</mi></math>. Clearly, every limited set in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m8\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> is a Dunford-Pettis set, but the converse is not true in general"], "text": ["Let us recall that a bounded subset A of X is called a Dunford-Pettis set (resp. a limited set) in X if each weakly null sequence in X* (resp. weak* null sequence in X*) converges uniformly to zero on A. Clearly, every limited set in X is a Dunford-Pettis set, but the converse is not true in general"], "count": ["1"], "color": ["#EDBDBC"], "csrfmiddlewaretoken": ["{{ csrf_token }}"]}
{"parentHTMLfraction": ["<p class=\"ltx_p\"><div style=\"background-color: rgb(237, 189, 188);\">Let us recall that a norm bounded subset <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m1\" class=\"ltx_Math\" alttext=\"A\" display=\"inline\"><mi>A</mi></math> of <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m2\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> is called a <em class=\"ltx_emph ltx_font_italic\">Dunford-Pettis set</em> (resp.a <em class=\"ltx_emph ltx_font_italic\">limited set</em>) if each weakly null sequence\nin <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m3\" class=\"ltx_Math\" alttext=\"X^{\\ast}\" display=\"inline\"><msup><mi>X</mi><mo>∗</mo></msup></math> (resp. weak* null sequence in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m4\" class=\"ltx_Math\" alttext=\"X^{\\ast}\" display=\"inline\"><msup><mi>X</mi><mo>∗</mo></msup></math>) converges\nuniformly to zero on <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m5\" class=\"ltx_Math\" alttext=\"A\" display=\"inline\"><mi>A</mi></math>.</div> An operator <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m6\" class=\"ltx_Math\" alttext=\"T:X\\rightarrow Y\" display=\"inline\"><mrow><mi>T</mi><mo>:</mo><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></mrow></math> is called <em class=\"ltx_emph ltx_font_italic\">Dunford-Pettis</em> if <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m7\" class=\"ltx_Math\" alttext=\"x_{n}\\overset{w}{\\rightarrow}0\" display=\"inline\"><mrow><msub><mi>x</mi><mi>n</mi></msub><mo>⁢</mo><mover accent=\"true\"><mo>→</mo><mo>𝑤</mo></mover><mo>⁢</mo><mn>0</mn></mrow></math> in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m8\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> implies <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m9\" class=\"ltx_Math\" alttext=\"\\left\\|Tx_{n}\\right\\|\\rightarrow 0\" display=\"inline\"><mrow><mrow><mo>∥</mo><mrow><mi>T</mi><mo>⁢</mo><msub><mi>x</mi><mi>n</mi></msub></mrow><mo>∥</mo></mrow><mo>→</mo><mn>0</mn></mrow></math>, equivalently, if <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m10\" class=\"ltx_Math\" alttext=\"T\" display=\"inline\"><mi>T</mi></math> carries\nrelatively weakly compact subsets of <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m11\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> onto relatively compact subsets of <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m12\" class=\"ltx_Math\" alttext=\"Y\" display=\"inline\"><mi>Y</mi></math>. An operator <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m13\" class=\"ltx_Math\" alttext=\"T:X\\rightarrow Y\" display=\"inline\"><mrow><mi>T</mi><mo>:</mo><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow></mrow></math> is said to be <em class=\"ltx_emph ltx_font_italic\">limited</em> whenever <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m14\" class=\"ltx_Math\" alttext=\"T\\left(B_{X}\\right)\" display=\"inline\"><mrow><mi>T</mi><mo>⁢</mo><mrow><mo>(</mo><msub><mi>B</mi><mi>X</mi></msub><mo>)</mo></mrow></mrow></math> is a limited set in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m15\" class=\"ltx_Math\" alttext=\"Y\" display=\"inline\"><mi>Y</mi></math>, equivalently, whenever <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m16\" class=\"ltx_Math\" alttext=\"\\left\\|T^{\\ast}\\left(f_{n}\\right)\\right\\|\\rightarrow 0\" display=\"inline\"><mrow><mrow><mo>∥</mo><mrow><msup><mi>T</mi><mo>∗</mo></msup><mo>⁢</mo><mrow><mo>(</mo><msub><mi>f</mi><mi>n</mi></msub><mo>)</mo></mrow></mrow><mo>∥</mo></mrow><mo>→</mo><mn>0</mn></mrow></math> for every\nweak* null sequence <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m17\" class=\"ltx_Math\" alttext=\"\\left(f_{n}\\right)\\subset Y^{\\ast}\" display=\"inline\"><mrow><mrow><mo>(</mo><msub><mi>f</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>⊂</mo><msup><mi>Y</mi><mo>∗</mo></msup></mrow></math>.</p>"], "htmlFraction": ["Let us recall that a norm bounded subset <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m1\" class=\"ltx_Math\" alttext=\"A\" display=\"inline\"><mi>A</mi></math> of <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m2\" class=\"ltx_Math\" alttext=\"X\" display=\"inline\"><mi>X</mi></math> is called a <em class=\"ltx_emph ltx_font_italic\">Dunford-Pettis set</em> (resp.a <em class=\"ltx_emph ltx_font_italic\">limited set</em>) if each weakly null sequence\nin <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m3\" class=\"ltx_Math\" alttext=\"X^{\\ast}\" display=\"inline\"><msup><mi>X</mi><mo>∗</mo></msup></math> (resp. weak* null sequence in <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m4\" class=\"ltx_Math\" alttext=\"X^{\\ast}\" display=\"inline\"><msup><mi>X</mi><mo>∗</mo></msup></math>) converges\nuniformly to zero on <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"S1.p2.m5\" class=\"ltx_Math\" alttext=\"A\" display=\"inline\"><mi>A</mi></math>."], "text": ["Let us recall that a norm bounded subset A of X is called a Dunford-Pettis set (resp.a limited set) if each weakly null sequence in X∗ (resp. weak* null sequence in X∗) converges uniformly to zero on A."], "count": ["2"], "color": ["#EDBDBC"], "csrfmiddlewaretoken": ["{{ csrf_token }}"]}