By Garrett P.
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Such an assertion is equivalent to various completeness hypotheses, which we will investigate later. 14. Appendix: Urysohn and density of Co Urysohn’s lemma is the technical point that allows us to relate measurable functions to continuous functions. Then, from the Lebesgue definition of integral, we can prove the density of continuous functions in L 2 spaces, for example. Theorem: (Urysohn) Let X be a locally compact Hausdorff topological space. In X, given a compact subset K contained in an open set U , there is a continuous function 0 ≤ f ≤ 1 which is 1 on K and 0 off U .
That is, there is indeed a local basis of convex balanced opens at 0. For the triangle inequality for pU , given v, w ∈ V , let t1 , t2 be such that v ∈ t · U for t ≥ t1 and w ∈ t · U for t ≥ t2 . Then, using the convexity, v + w ∈ t1 · U + t2 · U = (t1 + t2 ) · t2 t1 ·U + ·U t1 + t 2 t1 + t 2 ⊂ (t1 + t2 ) · U This gives the triangle inequality pU (v + w) ≤ pU (v) + pU (w) Finally, we should check that the semi-norm topology is exactly the original one. This is unsurprising. It suffices to check at 0.
Claim: Products and limits of topological vector spaces exist. In particular, limits are closed (linear) subspaces of the corresponding products. If the factors or limitands are locally convex, then so is the product or limit. Remark: Part of the point is that products and limits of locally convex topological vector spaces in the larger category of not-necessarily locally convex topological vector spaces are nevertheless locally convex. That is, enlarging the category in which we take test objects does not change the outcome, in this case.