Question

Difference between topological space and vector space

Answer 1

Answer:

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space (an algebraic structure) which is also a topological space, the latter thereby admitting a notion of continuity. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

The elements of topological vector spaces are typically functions or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions.

Hilbert spaces and Banach spaces are well-known examples.

Unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers C or the real numbers R.

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