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The Stone-Weierstrass theorem has an analog for the algebras of smooth functions, called Naсhbin's theorem: An involutive subalgebra $A$ in the algebra ${\mathcal C}^\infty(M)$ of smooth functio …

I asked this at mathstackexchange a week ago, without success. I think the Gelfand–Naimark–Segal construction must be a functor in some sense, but I can't find an explicit statement anywhere. Can an …

Let $\mu$ be a finite positive measure on a set $M$: $$ \mu(M)<\infty. $$ As is known, the Banach dual space $L_\infty(\mu)^*$ to the space $L_\infty(\mu)$ contains $L_1(\mu)$, but (excluding some tri …

Harmonic analysis is concentrated mostly on studying locally compact groups. I would like to ask people about examples of non-locally compact topological groups that are interesting in connection with …

I strongly recommend you to read the François Bruhat paper, that Osborne cites. For an arbitrary locally compact (not necessarily abelian) group $G$ Bruhat defines smooth function $\varphi:G\to{\mathb …

Let $X$ be a Banach space and $B(X)$ be its space of all (bounded) operators. A nuclear functional on $B(X)$ is a linear functional $u:B(X)\to{\mathbb C}$ that can be represented in the form $$ u(A)=\ …

The Paley-Wiener theorem as it is presented in W.Rudin's "Functional analysis" (Theorem 7.23), will it be satisfactory for you? In your situation: $\mu$ has compact support if and only if $F_\mu$ can …

You can take a look at the book by A.N.Kolmogorov and S.V.Fomin Elements of the theory of functions and functional analysis. They discuss measure theory and its connection with functional analysis (an …

Arnold, if $l$ is not obliged to belong to $\mathbb N$, i.e. the index set for $l$ can be arbitrary (for $l\in\mathbb N$ this seems to be also true, but this requires a verification), then the usual ( …

Let $A$ and $B$ be topological associative algebras (no matter, in which sense, for example, over $\mathbb C$, with identity, and with separately continuous multiplication). Let us say that a (continu …

For each pair of functions $x,y\in L_2[0,1]$ let us denote by $x\cdot y$ their pointwise product $$ (x\cdot y)(t)=x(t)\cdot y(t),\quad t\in [0,1]. $$ It belongs to $L_1[0,1]$ due to the Cauchy-Bunyako …

I suppose, you mean the usual topology on $D({\mathbb R}^n)$ defined for example in Rudin's book. Take $D_N=\{\varphi\in D({\mathbb R}^n):\ {\rm supp}\varphi\subseteq\{x\in{\mathbb R}^n:\ |x|\le N\} \ …

Recently a new application of functional analysis in geometry appeared, the study of envelopes of topological algebras. It allows to look at "big" geometric disciplines -- complex geometry, differenti …

Let $X$ and $Y$ be locally convex spaces, and $\varphi: X\to Y$ a linear continuous mapping. Suppose first that $S$ is a compact set in $X$. Then $\varphi$, being considered as a mapping from $S$ to $ …

I asked this a week ago at math.stackexchange, but without success. As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "L …