View source for Locally convex topology

A (not necessarily Hausdorff) topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l0603701.png" /> on a real or complex [[Topological vector space|topological vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l0603702.png" /> that has a basis consisting of convex sets and is such that the linear operations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l0603703.png" /> are continuous with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l0603704.png" />. A locally convex topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l0603705.png" /> on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l0603706.png" /> is defined analytically by a family of semi-norms (cf. [[Semi-norm|Semi-norm]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l0603707.png" /> as the topology with basis of neighbourhoods of zero consisting of the sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l0603708.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l0603709.png" /> runs through the natural numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037010.png" /> is the family of all finite intersections of the sets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037012.png" />; such a family of semi-norms is said to be a generator for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037013.png" /> or to generate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037014.png" />. The topology induced by a given locally convex topology on a vector subspace, the quotient topology on a quotient space and the topology of a product of locally convex topologies, are also locally convex topologies. A topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037015.png" /> on a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037016.png" /> is a locally convex topology if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037017.png" /> is the topology of [[Uniform convergence|uniform convergence]] on equicontinuous subsets of the [[Adjoint space|adjoint space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037018.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037021.png" />, be vector spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037022.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037023.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037024.png" /> (respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037025.png" />) be a linear mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037026.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037027.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037028.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037029.png" />) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037030.png" /> be a locally convex topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037032.png" />. The weakest topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037033.png" /> for which all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037034.png" /> are continuous mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037035.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037036.png" /> is called the projective topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037037.png" /> with respect to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037038.png" />. The projective topology is a locally convex topology. In particular, the least upper bound of a family of locally convex topologies on a given vector space, the induced topology on a subspace and the topology of a product of locally convex topologies are projective topologies (and therefore locally convex topologies). The strongest locally convex topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037039.png" /> with respect to which all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037041.png" />, are continuous mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037042.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037043.png" /> is called the inductive topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037044.png" /> with respect to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060370/l06037045.png" />. In particular, the quotient topology of a given locally convex topology and the topology of a direct sum of locally convex topologies are inductive topologies (and therefore locally convex topologies). The concepts of projective and inductive locally convex topologies make it possible to define the operations of projective and inductive limits in the category of locally convex spaces and their linear mappings.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Topological vector spaces" , Addison-Wesley  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Macmillan  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.V. Kantorovich,   G.P. Akilov,   "Functional analysis" , Pergamon  (1982)  (Translated from Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,   "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Jarchow,   "Locally convex spaces" , Teubner  (1981)  (Translated from German)</TD></TR></table>