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Results tagged with banach-spaces
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user 75422
A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.
1
vote
Weak continuity of a vector valued function
"Weakly continuous almost everywhere" means there is a negligible set of $t$'s outside which $t\mapsto x^*(f(t))$ is continuous for all $x^*\in\ell_\infty^*$, not the other way round. But $\delta_s(f( …
- 3,085
1
vote
"Generalisation" of one-parameter semigroups
Upon reflection, I find this approach interesting enough, although the example you gave is too elementary to be sure.
Provided your $K_G$ contains $\mathcal D_0:=C_c^\infty(0,\infty)$, your "bilatera …
- 3,085
0
votes
Completion of $\mathcal{S}(\mathbb{R})$ for a given norm
In the general case of a normed linear space $X$ and a larger quasi-complete Hausdorff topological vector space $E$ (as $\mathcal S'$ is) with $X\subseteq E$ : it can be completed within $E$ iff its u …
- 3,085
12
votes
Accepted
Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\D...
"The" completion is not always a space of functions, for $N=1$ or $2$ for example it is a quotient $D^{-2}L^2/P_1$ (equivalence classes of functions $u\in H^2_{loc}$ with $\partial_i \partial_j u\in L …
- 3,085