View source for 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 $E$ 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 $E$ 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Banach,   H. Steinhaus,   "Sur le principe de la condensation de singularités"  ''Fund. Math.'' , '''9'''  (1927)  pp. 50–61</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Topological vector spaces" , Addison-Wesley  (1977)  (Translated from French)  {{MR|0583191}} {{ZBL|1106.46003}} {{ZBL|1115.46002}} {{ZBL|0622.46001}} {{ZBL|0482.46001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Macmillan  (1966)  {{MR|0193469}} {{ZBL|0141.30503}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,   "Topological vector spaces" , '''1''' , Springer  (1969)  {{MR|0248498}} {{MR|0178335}} {{ZBL|0179.17001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Kelley,   I. Namioka,   "Linear topological spaces" , Springer  (1963)  {{MR|0166578}} {{ZBL|0115.09902}} </TD></TR></table>