locally convex space in a sentence

As this property is very useful in functional analysis, generalizations of normed vector spaces with this property are studied under the name locally convex spaces.
This gives an identification of real-valued Radon measures with the dual space of the locally convex space \ mathcal { K } ( X ).
Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.
For arbitrary locally convex space X the spaces X ^ { \ vartriangle \ triangledown } and X ^ { \ triangledown \ vartriangle } are stereotype.
Silva worked in analytic functionals, the theory of distributions, vector-valued distributions, ultradistributions, the operational calculus, and differential calculus in locally convex spaces.

Each locally convex space X can be transformed into a stereotype space with the help of the standard operations of pseudocompletion and pseudosaturation defined by the following two propositions.
Using the bilinear map, semi norms can be constructed to define a polar topology on the vector spaces and turn them into locally convex spaces, generalizations of normed vector spaces.
I'm still struggling with it, but when I've got it all internalized, I'm going to add a section to locally convex spaces about it.
On the other hand, the Hahn Banach theorem, which applies to all locally convex spaces, guarantees the existence of many continuous linear functionals, and so a large dual space.
A notable example of a result which had to wait for the development and dissemination of general locally convex spaces ( amongst other notions and results, like unit ball of the dual is metrizable ).

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