Suppose that X, Y, and Z are locally convex spaces and let \ mathcal { G }'and \ mathcal { H }'be the collections of equicontinuous subsets of X ^ * and Y ^ *, respectively.
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Given locally convex spaces and with families of seminorms " ? " } } and " ? " } } respectively, a linear map is continuous if and only if for every, there exist and such that for all in
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The pseudosaturation of X can be imagined as the " nearest to X from the inside " pseudosaturated locally convex space, so that the operation X \ mapsto X ^ \ vartriangle strengthen the topology of X, but does not change the elements of X.
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Given a barrelled space " X " and a locally convex space " Y ", then any family of pointwise bounded continuous linear mappings from " X " to " Y " is equicontinuous ( even uniformly equicontinuous ).
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*The Krein Milman theorem states that if " S " is convex and compact in a locally convex space, then " S " is the closed convex hull of its extreme points : In particular, such a set has extreme points.
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One can imagine the pseudocompletion of X as the " nearest to X from the outside " pseudocomplete locally convex space, so that the operation X \ mapsto X ^ \ triangledown adds to X some supplementary elements, but does not change the topology of X ( like the usual operation of completion ).
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