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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 asked this question in stackexchange, but it flashed and disappeared: Let $A$ be a C*-algebra and $\exp(x)=\sum_{n=0}\frac{x^n}{n!}$, the usual exponential function from $A$ into $A$. Is it true th …

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 …

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 …

I am not a specialist in measure theory, so excuse me if this is simple. Let $\mu$ be a finite measure on a set $X$ (for example, the Lebesgue measure on $[0,1]$). Integrable functions on $X$ can be …

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 ( …

A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality $$ L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu) $$ for measures $\mu$ and $\nu$. At the same …

Let $X$ be a compact Hausdorff topological set, and $Y$ be its closed subset. Is the ideal of functions vanishing on $Y$ $$ I=\{f\in C(X):\ \forall y\in Y\ f(y)=0\} $$ complementable (as a closed subs …

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 …

François Bruhat in his paper of 1961 (see page 61) gave a definition (and proved some properties) of the space ${\mathcal S}(G)$ of the Schwartz test functions on an arbitrary abelian locally compact …

As is known (see Kadison-Ringrose, 3.4.1) each closed ideal $I$ in the $C^*$-algebra $C(X)$ of continuous functions on a compact space $X$ has the form $$ I=\{f\in C(X): \ \forall x\in S\quad f(x)=0 \ …

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 …

There are four most important functional spaces in analysis: the space $\mathcal{C}(M)$ of continuous functions on a topological space, the space $\mathcal{E}(M)$ of smooth functions on a smooth man …

Let $X$ be a locally convex space, $T$ a balanced convex compact set in $X$, and $f:X\to\mathbb{C}$ a linear functional which is (not necessarily continuous on $X$, but) continuous on $T$. It is not d …

In stereotype theory there is an isomorphism of stereotype spaces (or, what is the same here, an isomorphism of locally convex spaces) $$ {\mathcal C}^\star(X)\circledast {\mathcal C}^\star(Y)\cong{\m …