Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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singular values of compact operators acting on Hilbert spaces

Let $A$ be a bounded linear (compact) operator acting op a separable Hilbert space $H$, and let $\varphi_1,\varphi_2,\ldots$ be an orthonormal basis of $H$. I Assume that $|\left< A\varphi_1,\...
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Solve functional equation: f(x+y)=f(x)+f(y) [on hold]

Solve following functional equation: $$f:R\to R$$ $$f(x+y)=f(x)+f(y)$$
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11 views

The discrete Laplacian

I am working on the $d$-dimensional integer lattice. Let $S$ be a random walk with increment distribution $p$. Given the distribution $p$ we can define the discrete Laplacian just as in Wikipedia is ...
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1answer
14 views

Is there any approach to computer vision that doesn't make use of geometry?

I've long been interested in applying my background in functional analysis (especially wavelets) and other related areas to actually create something with "real world" value (not that I don't enjoy ...
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39 views

$||\hat{f} ||_{\infty} = \lim _ {n \rightarrow \infty} (||f^{(n)}||_1)^{1/n}$

Let $f \in L^2 \cap L^1$ on the Real line, and define $f^{(n)}$ to be the $n$-fold convolution $f \circ f ... \circ f $. I want to show that $||\hat{f} ||_{\infty} = \lim _ {n \rightarrow \infty} (||...
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Dirac functional embedding

I got the following statements to show. Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with ...
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1answer
15 views

Relationship between spectral rays commuting

Show sing binomial Newton that if $S$ and $T$ are continuous operators on Banach space $B$ such that $ST = TS$ then $$r_\sigma(T + S) \leq r_\sigma(T) + r_\sigma(S)$$ where $r_\sigma$ is spectral ...
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1answer
26 views

Showing that if $E^*$ is reflexive, then $E$ is reflexive , for a Banach space $E$. [duplicate]

Let $E$ be a banach space. I have already shown that if $E$ is reflexive then $E^{*}$ is reflexive. Now I want to show that if the dual space $E^{*}$ is reflexive, then $E$ is reflexive. If $E^...
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16 views

How can we extend $\operatorname{div}:C_c^\infty(\Omega)^d\to L^p(\Omega)$ to $W_0^{1,\:p}(\Omega)^d$?

Let $d\in\mathbb N$ and $p\ge 1$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open and $$W_0^{1,\:p}(\Omega):=\overline{C_c^\infty(\Omega)}^{\left\|\;\cdot\;\...
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2answers
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The form of a normal operator with only one element in its spectrum

Let be $H$ a Hilbert space. Show that if $T$ is a normal linear operator continuous (i.e. $T^*T = TT^*$, with $T^*$ the Hilbert adjunct of $T$) and your spectrum $\sigma(T) = \{\lambda\}$, than $T = \...
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1answer
27 views

Extension of Fourier transform to $L^2(\mathbb{R})$

We defined the fourier transform and it's inversion for the Schwartz class $S(\mathbb{R})$. Since $S(\mathbb{R})$ is dense in $L^2(\mathbb{R})$, we can find for a given $f\in L^2(\mathbb{R})$ a ...
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17 views

Definition of extrema in calculus of variations

I am reading Gelfand and Fomin about Calculus of Variations and in page 12 they say: ' Analogously, we say that the functional $J[y]$ has a (relative) extremum for $y=\hat{y}$ if $J[y]-J[\hat{y}]$ ...
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1answer
37 views

Prove that $\lim_{n\rightarrow \infty} T_n(x) = T(x) \iff \lim_{n\rightarrow \infty}\sup_{x \in K}\|T_n(x) - T(x)\| = 0$

Suppose $X$ and $Y$ are Banach spaces and $T:X \rightarrow Y$ is a BLO and $K$ is a compact subset of $X$. Prove that: $$\lim_{n\rightarrow \infty} T_n(x) = T(x) \iff \lim_{n\rightarrow \infty} \sup_{...
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1answer
22 views

Linear span of weighted powers

I am reading Functional Analysis by Peter Lax, and I do not understand the passage where it says that $w(t)e^{i\zeta t}$ belongs to $C$, where: $\zeta$ is a complex variable, and $C$ is the set of ...
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1answer
17 views

Relation between Ranges of compact operators

I am reviewing functional analysis and getting stuck in this problem. Let $X,Y$ be two Banach spaces and $A,B\in L(X,Y)$. Prove that if $A$ is a compact operator and $R(B)\subset R(A)$ then $B$ is ...
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23 views

Proof that group of invertible elements in a Banach algebra have 1 or infinite connected components?

I'm trying to reconcile this proof that I've read that a group of invertible elements in a commutative (complex) Banach algebra have 1 or infinite connected components with this example I'm looking at....
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2answers
43 views

Integral equation of the form $\int_{-\infty}^{\infty} e^{-a t^4} g(x,t) dt = e^{-b x^4}$

How to solve an integral equation of the following form \begin{align} \int_{-\infty}^{\infty} e^{-a t^4} g(x,t) dt = e^{-b x^4} \end{align} where $a$ and $b$ are some positive constants. I am not ...
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34 views

Prove $\int_\Omega f(x) \,dx=f(x_B) \int_\Omega1 dx+ \mathcal O(\int_\Omega1 dx \cdot \sup_{x,y\in\Omega}\|x-y\|_2^2)$?

Let $\Omega \subset \Bbb R^n$ be a convex domain and $f: \Omega \to \Bbb R $ and $f \in \mathcal C^2(\Omega)$. Let $x_B $ be the barycentre of $\Omega$ with $$x_B:= \frac{\int_\Omega x \,dx}{\int_\...
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1answer
33 views

Exercise 16 - chapter 11 - From Rudin's Functional Analysis

I'm trying to solve this problem, which comes from the book mentioned in the title. Suppose $A$ is a Banach algebra, $m$ is an integer, $m\geq2$, $K<\infty$, and $$\|x\|^m \leq K\|x^m\| $$ ...
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1answer
55 views

What is the mathematical meaning of a quantum operator mean?

(Context: I am learning functional analysis using the book by Erwin Kreyszig "Introductory functional analysis with applications." The last chapter is dedicated to the applications of functional ...
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34 views

Convergence in $L^2(\Bbb R)$ implies convergence of the norms [on hold]

If $||f_n-f||_{L^2(\mathbb{R})}\to 0$ is it always true that $||f||_{L^2(\mathbb{R})}=\lim_{n\to\infty}||f_n||_{L^2(\mathbb{R})}$?
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Let $H$ be hilbert and $T$ a BLO, such that $T:H\rightarrow H$. Prove that $\langle T(x),x \rangle = 0$ implies $T = 0$.

Let $H$ be hilbert and $T$ a BLO, such that $T:H\rightarrow H$. Prove that $\langle T(x),x \rangle = 0$ implies $T = 0$. Any hints to tackle this problem? i tried writing x as $x = u + v$ where $u \...
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1answer
15 views

Approximation of bounded Sobolev functions in $L^\infty$

I'm trying to understand the problem of my former question in detail, and the crucial point (at least in my attempts to solve the problem) seems to be the following: Let $\Omega\subset \mathbb R^d$...
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1answer
10 views

Semi-finite trace on a von Neumann algebra: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. Normality means that $\tau(x) = \sup_i \tau(x_i)$ if $x \in N_+$ is the limit ...
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Is strong operator topology space $(B(H), SOT)$ reflexive?

It is true that $(B(H), SOT)$ is semireflexive, in which $H$ is a Hilbert space, and $B(H)$ is the set of all bounded linear operators from $H$ to $H$ with strong operator topology. As a starting ...
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1answer
18 views

linear preserving norm prolongement problem [on hold]

consider ($\mathbb{R}^2, \|.\|_\infty)$ , where $\|(x,y)\|_\infty = \max\{|x|, |y|\}$. Let $f: \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x,y)=\frac{x+y}{2}$ and $g$ be the norm preserving linear ...
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Are all important function spaces vector spaces?

EDIT: I definitely agree with Mike Miller that the question as written originally/below is too general. Is everything an analyst could ever care about locally homeomorphic to a T1 topological ...
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Poincaré's inequality proof for $u \in W_0^{1,p}(\Omega)$.

I am trying to prove Poincare's inequality for $u \in W_0^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is an open bounded set and $1 \leq p < \infty$. This is Poincare's inequality: $||...
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+50

Linear transformation $T$ such that for every extension $\overline{T}$, $\|\overline{T}\|>\|T\|$.

Let $E$ and $F$ be normed spaces such that $\dim F < \infty$, $G$ a subspace of $E$ and $T:G\rightarrow F$ a continuous linear map. I know that there exists a continuous linear extension $\overline{...
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36 views

Hahn-Banach theorem, dual space

Let $(X, ||\cdot||_X)$ be a normed vector space and $(X^{\ast},||\cdot||_{X^{\ast}})$ its dual space. I have to proof, that $$ \forall x\in X:\quad ||x||_X = \sup_{T\in X^{\ast}}\{|T(x)| : ||T||_{X^{\...
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Let $T: Y \rightarrow X$ be a isometry, where $X$ is Banach and reflexive. Construct a completion $(Z,i)$, where $Z$ is banach and $i(Y)$ is dense

i am kind of new to these second duals and reflexives spaces and saw this question which i don't really understand. Can you help me a bit or hint me in the right direction? Let $T: Y \rightarrow X$ ...
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32 views

$W_0^{1,\:p}(\Lambda)$ is dense in $L^2(\Lambda)$

Let $d\in\mathbb N$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Lambda\subseteq\mathbb R^d$ be open with $\lambda(\Lambda)<\infty$ $p\ge 2$ $W^1(\Lambda)$ denote the set of weakly ...
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1answer
42 views

A Question Regarding Stone's Formula

Let $A$ be a bounded self-adjoint operator on a separable Hilbert space $\mathcal{H}$: $$ A\in\mathcal{B}\left(\mathcal{H}\right)\,,\,A=A^\ast$$ Stone's formula (Reed & Simon Theorem VII.13, as an ...
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Generating set of Baire sigma-algebra

I got the following statement to prove: Let $X$ locally compact and $\operatorname{Ba}(X)$ the Baire-$\sigma$-algebra, i. e. the smallest $\sigma$-algebra with respect to which all functions in $f \...
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1answer
21 views

Countable weighted shift has no invariant subspace.

Suppose I have $T(e_n)=w_ne_{n+1}$ where $w_n>0$ (and are bounded) and $\{e_n\}$ denotes the canonical basis of $l^{2}(\mathbb{N})$. I would like to prove that the only (closed) invariant ...
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1answer
22 views

One-sided smooth approximation of Sobolev functions

I'm currently trying to specialise a rather general variational inequality to known simple examples to check if my assumptions on the problem are plausible. While doing this, I stepped over the ...
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45 views

Solve the nth zero of a function. [on hold]

Say I have a mystery continuous function, could be anything. f(x) Assuming we don't know the distribution of the zeros of the function, Is there a known way to solve the nth zero (hits the x-axis)? ...
3
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1answer
77 views

Tricky norm-inequality $\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$ for $p \in (0,1)$

For $r>p \ge 1$ one can show that in $\mathbb{C}^n$ we have $$\|x\|_p \le n^{\frac{1}{p}- \frac{1}{r}} \|x\|_r.$$ My question is now: Does this also hold for $1 \ge r>p>0$? Obviously we ...
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1answer
50 views

Stone-Weierstrass theorem of $\mathbb{S}^2$

Someone told me that every continuous function on $\mathbb{S}^2$ could be expressed as a uniform limit of restrictions to $\mathbb{S}^2$ of polynomials. Does this result come from the Stone-...
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1answer
32 views

is every nonzero eigenvalue's eigenspace finite dimensional [on hold]

T is a bounded linear operator on a Banach space , for every non-zero eigenvalue a , is its eigenspace always finite-dimensional ?
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1answer
20 views

Definition of Frechet space

I have a question regarding the two equivalent definitions of a Frechet space (cf. Wikipedia): According to Def.1, a Frechet space is a topological VS $X$, such that $X$ is locally convex ($0\in X$ ...
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34 views

Dual of a vector space of continuous functions

Let $X=C_{\partial}([a,b])$ be the Banach space of continuous real-valued functions on $[a,b]\subset \mathbb{R}$ such that $f(a)=f(b)=0$ equipped with the supremum norm. I want to now what is its ...
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1answer
22 views

a problem concerning continuous functions of bounded variation [duplicate]

Here is a problem: Suppose $f,g: [a,b]\rightarrow \mathbb{R}$ are both continuous and of bounded variation. Show that the set $\{(f(t),g(t))\in\mathbb{R}^2: t\in [a,b]\}$ CANNOT cover the entire unit ...
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67 views

Name of a Sobolev/Poincare type inequality

It is a basic analysis exercise to verify that for $u \colon [0,1] \times [0,\infty) \to \mathbb R$, if $u(0,t) = 0$ for all $t$, then $$ ||u(\cdot,t)||_\infty \leq ||u_x(\cdot,t)||_2,$$ where the ...
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34 views

Equivalent definitions for weak/distribution convergence

We let $X$ be a compact metric space and consider $C(X)$ to be the space of all continuous functions on $X$. The dual space of $C(X)$ can be seen as the set of all signed borel measure on $X$. My ...
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1answer
42 views

Prob. 2, Sec. 2.8 in Kreyszig's functional analysis text: What is the norm of these bounded linear functionals on $C[a,b]$?

Let $C[a,b]$ denote the normed space of all the continuous (real or complex-valued) functions defined (and continuous!) on the closed interval $[a,b]$ on the real line, where $a, b \in \mathbb{R}$ and ...
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36 views

T/F: Properties of every non–trivial topological vector space over $\mathbb{R}$ or $\mathbb{C}$

Note: by "non–trivial" I mean "not discrete", which to the best of my knowledge is equivalent for a TVS over $\mathbb{R}$ or $\mathbb{C}$. Since any such space is over a local field, it is ...