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Needs:
Norms
Norm Metrics
Needed by:
None.
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Wikipedia

Topological Vector Spaces

Why

For any normed vector space, the vector operations are continuous. We can abstract this notion.

Definition

A topological vector space is triple whose first coordinate is a vector space, whose second coordinate is a topology on the field of the vector space and whose third coordinate is a topology on the set of vectors, such that the vector operations are continuous with respect to their product topologies.

Motivating example

Suppose $(V, \norm{\cdot })$ is a normed vector space. Then the vector operations are continuous with respect to the topology induced by the metric induced by the norm.

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