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[parent] locally convex topological vector space (Definition)

Definition Let $ V$ 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 $ V$ has a basis where each member is a convex set, then $ V$ is a locally convex topological vector space [1].

Though most vector spaces occurring in practice are locally convex, the spaces $ L^p$ for $ 0<p<1$ 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.



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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.
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AMS MSC46-00 (Functional analysis :: General reference works )
 46A03 (Functional analysis :: Topological linear spaces and related structures :: General theory of locally convex spaces)

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