locally convex topological vector space

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 isa convex set, then $V$ is a locally convex topologicalvector 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.

单词	LocallyConvexTopologicalVectorSpace
释义	locally convex topological vector space 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 isa convex set, then $V$ is a locally convex topologicalvector 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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