View source for 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 <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]]].

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" />.

====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>