Banach-Steinhaus theorem

(Redirected from Banach–Steinhaus theorem)

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