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locally convex topological vector space

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Definition Let be a topological vector space over a subfield of the complex numbers (usually taken to be $\mathbb{R}$ or $\mathbb{C}$ ). If the topology of has a basis where each member is a convex set, then is a locally convex topological vector space [1].

Though most vector spaces occurring in practice are locally convex, the spaces for are examples of spaces which are not locally convex.

References

1: G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.

"locally convex topological vector space" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Attachments:: topology of locally convex spaces is generated by seminorms (Theorem) by asteroid

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Cross-references: vector spaces, convex set, basis, topology, complex numbers, subfield, topological vector space
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This is version 6 of locally convex topological vector space, born on 2003-07-05, modified 2006-02-17.
Object id is 4424, canonical name is LocallyConvexTopologicalVectorSpace.
Accessed 4407 times total.

Classification:

AMS MSC:	46-00 (Functional analysis :: General reference works )
	46A03 (Functional analysis :: Topological linear spaces and related structures :: General theory of locally convex spaces)

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