Questions tagged [functional-analysis]

Question 1

Let $\{H_i\}_{i\in I}$ be a directed system of Hilbert spaces over a poset $I$ such that $H_i\subset H_j$ whenever $i\leq j$. We consider the direct limit $H=\cup_{i\in I} H_i$ and equip it with the ...

Question 2

Let $V$ be a normed vector space, $D\subseteq V$ and $\{f_n\}_{n\in\mathbb{N}}$ a sequence in $V^*$ such that for all $x\in D$, the sequence $\{f_n(x)\}_{n\in\mathbb{N}}$ is bounded. Is it true that ...

Question 3

Let $f_n$ be a sequence in $L^p(\mathbb{R})$ with $1 < p < \infty$ so that $f_n$ converges uniformly to $f$ on every compact subset of the real line. Find whether or not $f_n$ converges weakly ...

Question 4

I have been trying to prove for some time now the basic fact that if $A$ is a self-adjoint operator that has an empty essential spectrum then it has compact resolvent. Here I am using that \begin{...

Question 5

I want to prove that if $T:E \to E$ is a bounded operator on a Banach space $E$ and $\lambda \in \sigma_c (T)$, then there exists a sequence $x_n \in E$ such that $\| x_n \|=1$ and $Tx_n -\lambda x_n \...

Question 6

I am studying functional analysis using "Function Analysis An Introduction" by Yuli Eidelman, Vitali Milman, Antonis Tsolomitis. In page $44$ the following exercise (number $24$) is left for ...

Question 7

We know that the operator norm is $$\begin{align*} \sup\limits_{\|x\|\leq 1}\|Tx\| \end{align*}$$Then if $x_{0}\not= 0$ . Do we have $$\begin{align*} \sup\limits_{y \in \overline{B}(x_{0},1)} \|Ty\| \...

Question 8

More specifically, I'm considering the following problem: Let $v: \Omega \to \mathcal H$ be a Pettis-integrable Hilbert space-valued function defined on a perfect probability space $(\Omega, \mathcal ...

Question 9

In John B. Conway's "A Course in Functional Analysis" on pg 46. #6 says "Show that if $T: H \rightarrow K$ is a compact operator and $\{e_n\}$ is any orthonormal sequence in H, then $||...

Question 10

Suppose $f:\mathbb{R}\to\mathbb{R}$ is an explicit everywhere surjective function whose graph has Hausdorff dimension $2$ with a zero $2$-d Hausdorff measure. Since the integral of $f$ w.r.t. $2$-d ...

Question 11

Question: Let $X$ be a Hausdorff topological vector space. Let $K, L \subseteq X$ be absolutely convex compact subsets such that $K \subseteq \bigcup_{c \geq 0} c L$. Does there necessarily exist some ...

Question 12

I am trying to find a reference (if the result is even true at all) to the following claims: If $A,B$ are $C*$-algebras such that the enveloping von Neumann algebra of the minimal tensor product $(A \...

Question 13

I am trying to fill in the details of the following (already discussed in these posts for example and also in Rudin, Reed and Simon, etc) fact: If $\lambda \in \sigma(A)$ is isolated for $A$ self-...

Question 14

Question: Let $X$ be a Hausdorff topological vector space. Let $A \subseteq X$ be a closed subspace. Then the quotient $\newcommand\quotient[2]{{^{\Large #1}}/{_{\Large #2}}} \quotient{X}{A}$ is again ...

Question 15

In Reed-Simon chapter VIII, the following theorem is stated: Let $\{A_n\}_{n=1}^\infty, A$ be self-adjoint operators. Suppose that $\mathcal{D}$ is a common core for $A_n, A$. Then if $A_n\varphi\to A\...

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Questions tagged [functional-analysis]

Understanding bounded sets locally convex direct/inductive limit of topological vector spaces.

Does pointwise boundedness of continuous linear form in a subset of a normed space implies pointwise boundedness in the clousure? [duplicate]

Does uniform convergence on compact subsets of $\mathbb{R}$ imply weak convergence in $L_p(\mathbb{R})$?

Self-adjoint operators with empty essential spectrum have compact resolvent

If $\lambda$ is in the continuous spectrum then $T-\lambda I $ is not bounded below

For $f\in E^{\ast}$ If $\ker \left(f \right) \leq E$ is closed then $f$ is bounded. [duplicate]

Is the operator norm bounded by the sup of operator on any closed ball that is not centered at origin? [duplicate]

Relative weak compactness of $\{\frac{1}{\mathbb{P}(E)} {\rm Gelfand}-\int_E v d\mathbb{P} : E\in\mathcal{F}\}$

Compact operator condition

What is an applicable way of averaging an everywhere surjective function whose graph has Hausdorff dimension $2$ with zero $2$-d Hausdorff measure?

If $K,L$ are absolutely convex compact subsets with $K \subseteq \bigcup_{c \geq 0} c L$, does there exist some $c_0 \geq 0$ with $K \subseteq c_0 L$?

Reference Request: Normal extension of minimal tensor product of homomorphisms

Isolated point of spectrum of a self-adjoint operator and spectral projectors

Is a compact subspace of a quotient of a topological vector space contained in the image of a compact subspace of the original space?

A condition for strong resolvent convergence and the condition that $\mathcal{D}$ is a simultaneous core

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