Topological Vector Spaces

Topological Vector Space

A topological vector space is a vector space that is also a topological space, where the vector space operations (addition and scalar multiplication) are continuous with respect to the topology.

e.g. Banach spaces, Hilbert spaces, and Sobolev spaces are all examples of topological vector spaces.

Proposition

Suppose is a topological vector space. Fix some , then the translation map , is a homeomorphism. Therefore, any topological vector space is topologically homogeneous.

Proof Suppose is the addition. Then every translation map is a composition: where each map is continuous. So is continuous. Its inverse is , which is also continuous.

Locally Convex Space

A topological vector space is locally convex if it has a basis of its topology consisting of convex open subsets. Equivalently, has a neighborhood basis consisting of convex open subsets.

Complete Topological Space

A Cauchy net in a topological vector space is a net such that for every open neighborhood of , there exists such that for all , .
A topological vector space is complete if any Cauchy net converges to a point in the space.

Usually the completeness is defined with respect to a metric,

Theorem

For a topological vector space , is iff it is iff it is (Hausdorff).

References

• Maria Infusino, Topological Vector Spaces (2017).
• Dietmar Vogt, Lectures on Fréchet Spaces.
• Paul Garrett, Seminorms and locally convex spaces.