I read in Kolmogorov-Fomin’s (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq a<b\leq\infty$, is not equal to 0 [i.e. $\forall x\in(a,b)\quad f(x)\ne 0$, as Daniel, whom I deeply thank, explains in his answer] and satisfy the condition $|f(x)|\leq Ce^{-\delta|x|}$ with $\delta>0$, then […]

From Wikipedia: Given a topological vector space $(X,τ)$ over a field $F$, $S$ is called bounded if for every neighborhood $N$ of the zero vector there exists a scalar $α$ so that $$ S \subseteq \alpha N $$ with $$ \alpha N := \{ \alpha x \mid x \in N\}. $$ I was wondering if […]

The following is a Theorem of Conway’s operator theory: I can not understand how he proves it. I think $\phi(\text{ ball B(H)})$ is compact if $\phi(\text{ ball B(H)})$ is closed subset of compact set $X$, but why is $X$ compact?

Let $X$ be a compact space. Let us call a function $f:X\to {\mathbb C}$ universally integrable if it is integrable with respect to each regular Borel measure $\mu$ on $X$ (i.e. a positive functional on ${\mathcal C}(X)$, according to Riesz–Markov–Kakutani representation theorem). Let us denote by ${\mathcal U}(X)$ the space of all universally integrable functions […]

What is the difference between the algebraic and the topological dual of a topological vector space, such as for example the Euclidean space $\mathbb{H}$? I am interested in intuitive as well as in detailled technical answers.

Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if $\left \| x \right \|_Y \leq C \left \| x \right \|_X$ for some $C \geq 0$ (general definition of continuous inclusions in normed spaces) […]

We say that $f:[0,1]\to \mathbb{R}$ satisfies Lusin’s condition (N) provided $$m(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$ where $m$ stands for the Lebesgue measure on $\mathbb{R}$. I found in here the following definition. We say that $f:[0,1]\to X$ satisfies Lusin’s condition (N) provided $$\mathcal{H}^1(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$ where $X$ is a metric […]

This question is motivated by this other one. A classical result of linear algebra states what follows Up to scalar, trace is the only linear operator $\text{M}(n,k) \stackrel{t}{\to} k $ such that $t(AB) = t(BA)$. So there are two natural generalization of the problem. The first one is in the direction of replacing $k$ with […]

Suppose $X$ is a topological vector space and $X = V \oplus W$ is a decomposition of $X$ into closed subspaces. The decomposition gives rise to a projection $P$ onto $V$ (depending on the choice of complement $W$): \begin{equation} P:X \rightarrow X \\v+w \rightarrow v\end{equation} Is $P$ continuous? It seems like it ought to be […]

I’m working through Rudin’s Functional Analysis, and I am confused by a step in his proof for Theorem 1.24, which states that if X is a topological vector space with a countable local base, then there is a metric d on X such that a) d is compatible with the topology of X b) the […]

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