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Sep
28
comment Grothendieck on Topological Vector Spaces
I forgot the category of convenient spaces: ncatlab.org/nlab/show/convenient+vector+space. It seems to me it possesses some of these properties (without auto-duality, as far as I know). @PeterMichor can clarify this.
Sep
28
comment Grothendieck on Topological Vector Spaces
As to $\ell_1$, this is a question of habits. What we see every day becomes our own, like our relatives or friends.
Sep
28
comment Grothendieck on Topological Vector Spaces
Yemon, no, it isn't. Because this duality generates a closed monoidal category ${\tt Ste}$ of stereotype spaces, the second one in Analysis after the category ${\tt Ban}$ of Banach spaces with this property, and this auto-duality (with many other remarkable properties) makes ${\tt Ste}$ better than ${\tt Ban}$. IMHO. :)
Sep
28
comment Grothendieck on Topological Vector Spaces
As an illustration: who knows that every Banach space $X$ becomes relexive, $X^{\star\star}=X$, if we endow its dual space $X^\star$ with the compact-open topology? This was found in 1952 by Marianne Smith. When I am telling this to people they are surprised. Formally this is absurd: the simple explanation is less known than the intricate one. A reference for those who find this unexpected: en.wikipedia.org/wiki/Stereotype_space.
Sep
28
comment Grothendieck on Topological Vector Spaces
I upvoted Yemon's comment, but this was a mistake. In my opinion, the theory of TVS is indeed dead, and the most part of the guilt for this lies on Alexander Grothedieck. It must have been evident from the very beginning that there is something wrong in this abundance of topologies on the dual space, duality theories, counter-examples, etc. After its birth the theory immediately turned into a long list of counterexamples. The scientific explanation can't be so intricate, knotty, this is an abuse of professional knowledge.
Sep
28
comment Tangent space of the Fourier algebra $A(G)$
Thank you, Yemon!
Sep
27
accepted Tangent space of the Fourier algebra $A(G)$
Sep
27
comment Tangent space of the Fourier algebra $A(G)$
I thought, when $G$ is not discrete there must be at least usual derivatives $\frac{\partial}{\partial x_i}$... They are not continuous here?
Sep
27
comment Tangent space of the Fourier algebra $A(G)$
Yemon, how can this be? There are non-smooth functions in $A(G)$?
Sep
26
asked Tangent space of the Fourier algebra $A(G)$
Sep
25
comment A generalization of real characters on a group
Neil, but I suppose you mean $A[t]/\{t^{n+1}\}$, the quotient algebra.
Sep
25
revised A generalization of real characters on a group
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Sep
25
revised A generalization of real characters on a group
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Sep
24
comment A generalization of real characters on a group
@NeilStrickland, yes, excuse me for the mistake in the second condition! I corrected this.
Sep
24
revised A generalization of real characters on a group
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Sep
24
revised A generalization of real characters on a group
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Sep
24
revised A generalization of real characters on a group
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Sep
24
comment A generalization of real characters on a group
@YCor, yes initially $G$ and $A$ both were algebras. I reformulated this for the case when $G$ is a group.
Sep
24
asked A generalization of real characters on a group
Sep
24
accepted When is the Fourier algebra $A(G)$ enough close to the Fourier-Stieltjes algebra $B(G)$?