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 
and 
be locally convex linear topological spaces, where 
is a barrelled space, or let 
and 
be linear topological spaces, where 
is a Baire space. The following propositions are then valid. 1) Any subset of the set 
of continuous linear mappings of 
into 
which is bounded in the topology of simple convergence is equicontinuous (the uniform boundedness principle); 2) If a filter 
in 
contains a set bounded in the topology of simple convergence, and converges in the topology of simple convergence to some mapping 
of 
into 
, then 
is a continuous linear mapping of 
into 
, and 
converges uniformly to 
on each compact subset of 
[2], [3].
These general results make it possible to render the classical results of S. Banach and H. Steinhaus [1] more precise: Let 
and 
be Banach spaces and let 
be a subset of the second category in 
. Then, 1) if 
and 
is finite for all 
, then 
; 2) if 
is a sequence of continuous linear mappings of 
into 
, and if the sequence 
converges in 
for all 
, then 
converges uniformly on any compact subset of 
to a continuous linear mapping 
of 
into 
.
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) |
| [3] | H.H. Schaefer, "Topological vector spaces" , Macmillan (1966) |
Comments
References
| [a1] | G. Köthe, "Topological vector spaces" , 1 , Springer (1969) |
| [a2] | J.L. Kelley, I. Namioka, "Linear topological spaces" , Springer (1963) |
Banach-Steinhaus theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach-Steinhaus_theorem&oldid=28153