Locally convex topology

A (not necessarily Hausdorff) topology $ \tau $ on a real or complex topological vector space $ E $ that has a basis consisting of convex sets and is such that the linear operations in $ E $ are continuous with respect to $ \tau $. A locally convex topology $ \tau $ on a vector space $ E $ is defined analytically by a family of semi-norms (cf. Semi-norm) $ \{ {p _ \alpha } : {\alpha \in A } \} $ as the topology with basis of neighbourhoods of zero consisting of the sets of the form $ \{ n ^ {- 1} U \} $, where $ n $ runs through the natural numbers and $ U $ is the family of all finite intersections of the sets of the form $ \{ {x \in E } : {p _ \alpha ( x) \leq 1 } \} $, $ \alpha \in A $; such a family of semi-norms is said to be a generator for $ \tau $ or to generate $ \tau $. 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 $ \tau $ on a topological vector space $ E $ is a locally convex topology if and only if $ \tau $ is the topology of uniform convergence on equicontinuous subsets of the adjoint space $ E ^ {*} $.

Let $ E $ and $ E _ \alpha $, $ \alpha \in A $, be vector spaces over $ \mathbf R $ or $ \mathbf C $, let $ f _ \alpha $ (respectively $ g _ \alpha $) be a linear mapping of $ E $ into $ E _ \alpha $ (respectively, of $ E _ \alpha $ into $ E $) and let $ \tau _ \alpha $ be a locally convex topology on $ E _ \alpha $, $ \alpha \in A $. The weakest topology on $ E $ for which all $ f _ \alpha $ are continuous mappings of $ E $ into $ ( E _ \alpha , \tau _ \alpha ) $ is called the projective topology on $ E $ with respect to the family $ \{ {( E _ \alpha , \tau _ \alpha , f _ \alpha ) } : {\alpha \in A } \} $. 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 $ E $ with respect to which all $ g _ \alpha $, $ \alpha \in A $, are continuous mappings of $ ( E _ \alpha , \tau _ \alpha ) $ into $ E $ is called the inductive topology on $ E $ with respect to the family $ \{ {( E _ \alpha , \tau _ \alpha , g _ \alpha ) } : {\alpha \in A } \} $. 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

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

References

[a1]	G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
[a2]	H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)