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Difference between revisions of "Banach-Steinhaus theorem"

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A general appellation for several results concerning the linear-topological properties of the space of continuous linear mappings of one linear topological space into another. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b0152001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b0152002.png" /> be locally convex linear topological spaces, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b0152003.png" /> is a [[Barrelled space|barrelled space]], or let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b0152004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b0152005.png" /> be linear topological spaces, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b0152006.png" /> is a [[Baire space|Baire space]]. The following propositions are then valid. 1) Any subset of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b0152007.png" /> of continuous linear mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b0152008.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b0152009.png" /> which is bounded in the topology of simple convergence is equicontinuous (the uniform boundedness principle); 2) If a filter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520011.png" /> contains a set bounded in the topology of simple convergence, and converges in the topology of simple convergence to some mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520013.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520015.png" /> is a continuous linear mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520016.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520017.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520018.png" /> converges uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520019.png" /> on each compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520020.png" /> [[#References|[2]]], [[#References|[3]]].
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A general appellation for several results concerning the linear-topological properties of the space of continuous linear mappings of one linear topological space into another. Let $E$ and $F$ be locally convex linear topological spaces, where $E$ is a [[Barrelled space|barrelled space]], or let $E$ and $F$ be linear topological spaces, where $E$ is a [[Baire space|Baire space]]. The following propositions are then valid. 1) Any subset of the set $L(E,F)$ of continuous linear mappings of $E$ into $F$ which is bounded in the topology of simple convergence is equicontinuous (the uniform boundedness principle); 2) If a filter $P$ in $L(E,F)$ contains a set bounded in the topology of simple convergence, and converges in the topology of simple convergence to some mapping $v$ of $E$ into $F$, then $v$ is a continuous linear mapping of $E$ into $F$, and $P$ converges uniformly to $v$ on each compact subset of $E$ [[#References|[2]]], [[#References|[3]]].
  
These general results make it possible to render the classical results of S. Banach and H. Steinhaus [[#References|[1]]] more precise: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520022.png" /> be Banach spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520023.png" /> be a subset of the second category in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520024.png" />. Then, 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520026.png" /> is finite for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520028.png" />; 2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520029.png" /> is a sequence of continuous linear mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520030.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520031.png" />, and if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520032.png" /> converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520033.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520035.png" /> converges uniformly on any compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520036.png" /> to a continuous linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520038.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015200/b01520039.png" />.
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These general results make it possible to render the classical results of S. Banach and H. Steinhaus [[#References|[1]]] more precise: Let $E$ and $F$ be Banach spaces and let $M$ be a subset of the second category in $E$. Then, 1) if $H\subset L(E,F)$ and $\sup\{\|u(x)\|\colon u\in H\}$ is finite for all $x\in M$, then $\sup\{\|u\|\colon u\in H\}<\infty$; 2) if $u_n$ is a sequence of continuous linear mappings of $E$ into $F$, and if the sequence $u_n(x)$ converges in $F$ for all $x\in M$, then $u_n$ converges uniformly on any compact subset of $E$ to a continuous linear mapping $v$ of $E$ into $F$.
  
 
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Latest revision as of 11:05, 19 August 2014

A general appellation for several results concerning the linear-topological properties of the space of continuous linear mappings of one linear topological space into another. Let $E$ and $F$ be locally convex linear topological spaces, where $E$ is a barrelled space, or let $E$ and $F$ be linear topological spaces, where $E$ is a Baire space. The following propositions are then valid. 1) Any subset of the set $L(E,F)$ of continuous linear mappings of $E$ into $F$ which is bounded in the topology of simple convergence is equicontinuous (the uniform boundedness principle); 2) If a filter $P$ in $L(E,F)$ contains a set bounded in the topology of simple convergence, and converges in the topology of simple convergence to some mapping $v$ of $E$ into $F$, then $v$ is a continuous linear mapping of $E$ into $F$, and $P$ converges uniformly to $v$ on each compact subset of $E$ [2], [3].

These general results make it possible to render the classical results of S. Banach and H. Steinhaus [1] more precise: Let $E$ and $F$ be Banach spaces and let $M$ be a subset of the second category in $E$. Then, 1) if $H\subset L(E,F)$ and $\sup\{\|u(x)\|\colon u\in H\}$ is finite for all $x\in M$, then $\sup\{\|u\|\colon u\in H\}<\infty$; 2) if $u_n$ is a sequence of continuous linear mappings of $E$ into $F$, and if the sequence $u_n(x)$ converges in $F$ for all $x\in M$, then $u_n$ converges uniformly on any compact subset of $E$ to a continuous linear mapping $v$ of $E$ into $F$.

References

[1] S. Banach, H. Steinhaus, "Sur le principe de la condensation de singularités" Fund. Math. , 9 (1927) pp. 50–61
[2] N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French) MR0583191 Zbl 1106.46003 Zbl 1115.46002 Zbl 0622.46001 Zbl 0482.46001
[3] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966) MR0193469 Zbl 0141.30503


Comments

References

[a1] G. Köthe, "Topological vector spaces" , 1 , Springer (1969) MR0248498 MR0178335 Zbl 0179.17001
[a2] J.L. Kelley, I. Namioka, "Linear topological spaces" , Springer (1963) MR0166578 Zbl 0115.09902
How to Cite This Entry:
Banach-Steinhaus theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach-Steinhaus_theorem&oldid=32996
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article