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comment Topology in space of test functions $\mathcal{D}(\Omega)$ and space of distributions $\mathcal{D}'(\Omega)$
For every locally convex space $(X,\mathscr T)$ the weak topology $\sigma(X,X')$ (given by the directed family of seminorms $\max\lbrace |u(x)|. u\in E\rbrace$ with $E\subseteq X'$ finite) is coarser than $\mathscr T$ and gives the same dual space as well as the same bounded sets. For Montel spaces $(X,\mathscr T)$ this implies that both topologies have the same convergent sequences. $\sigma(X',X)$ is strictly coarser whenever $(X,\mathscr T)$ has a continuous norm (not only seminorm). All this has nothing to do with the particular spaces of distribution theory.
Feb
8
comment Is the module action $M\times M^*\to M^*$ jointly continuous?
@AliBagheri If this solvess your question why don't you accept Nick's answer?
Feb
4
comment Weak convergence in $L^2(0,T;X)$
Hence, Michael Renardy's comment applies.
Feb
2
comment Weak convergence in $L^2(0,T;X)$
According to the comment of Michael Renardy, you have to specify more explicitly what you mean by "weak convergence". Perhaps this should mean $L^2$-convergence of $\varphi \circ u_m$ for all $\varphi\in X'$?
Feb
2
comment Does bounded and closed equal compact for sets of Borel probability measures?
If $X$ is compact then the set of all Borel probability measures is weakly compact (and also weakly sequentially compact). This is a simple version of Prokhorov's theorem (or, in functional analytic terms, Alaoglu's theorem, functional analysts call the weak topology weak* topology). Of course, closed subsets of compact sets are again compact.
Feb
1
answered Description of $\big(\ell^\infty(\mathbb N)\big)^{\!*}$ via ultrafilters
Feb
1
awarded  Yearling
Jan
28
comment Extremally disconnected spaces and a measure theoretic property
Why is $U\setminus C$ clopen? It is open and this seems to be enough.
Jan
22
awarded  Popular Question
Jan
15
comment Why is the doubling dimension of any net of a metric space at most half of that of the metric space?
What's the relation between $N$ and $N_\epsilon$?
Jan
6
comment The decay rate of Hormander lemma is optimal or not?
Where is the dependence on $\lambda$ in the integral?
Dec
18
comment the double dual of “little l one” sequence space
But the standard notion then is $\ell_1$ or $\ell^1$.
Dec
17
comment the double dual of “little l one” sequence space
What is $\ell_1^\infty(\mathbb R)$?
Dec
17
revised Is $L_q(X^*)$ complemented in $(L_p(X))^*$?
Tag fa added
Dec
9
comment Stone-Weierstrass theorem for holomorphic functions?
If you are looking for an abstract condition it should apply to the concrete situation. I believe it is thus a good idea to try to formulate the concrete situation in general abstract terms.
Dec
9
comment Stone-Weierstrass theorem for holomorphic functions?
Can you formulate Runge's theorem in general terms of the algebra of rational functions with enough poles (one in each component of the complement)?
Dec
1
comment nontrivial theorems with trivial proofs
However, Levy's theorem that convergence in distribution is equivalent to convergence of the characteristic functions is less trivial.
Nov
29
comment Has every Lusin vector space a stronger Polish vector space topology?
A possible obstacle is the open mapping theorem: An ultrabornological l.c.s. does not have a strictly finer Frechet space topology. One would thus need some finer complete metrizable separable topology e.g. on $s'$, the space of sequences of at most polynomial growth.
Nov
23
comment When is the closed unit ball in a smaller Banach space closed in a larger Banach space?
I know the condition as a completeness lemma: If the unit ball of a normed space $X$ is closed in a Banach space $Y$ where $X$ is continuously embedded, then $X$ is also complete. A variant of this easy exercise to locally convex spaces is sometimes attributed to W. Robertson.
Nov
19
revised Completeness of nonharmonic Fourier Series
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