A Course on Topological Vector Spaces by Jürgen Voigt

Author:Jürgen Voigt
Language: eng
Format: epub
ISBN: 9783030329457
Publisher: Springer International Publishing

Proposition 9.4

Let E and F be topological vector spaces, E 0 ⊆ E a dense subspace, F Hausdorff and complete, and let u 0: E 0 → F be a continuous linear mapping. Then there exists a unique continuous extension u: E → F of u 0, and u is linear.

Proof

Note that F is regular, because the closed neighbourhoods of zero in F form a neighbourhood base of zero. Let be the neighbourhood filter of zero in E, and let . Then

is a filter in E 0 converging to x, hence a Cauchy filter. This implies that is a Cauchy filter base in F, hence convergent. Now Proposition 9.3 yields the existence and uniqueness of the continuous extension u of u 0.

In order to show the linearity of u we let and note that the set

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