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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1answer
11 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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0answers
14 views

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 ...
2
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0answers
18 views

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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106 views
+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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2answers
42 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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0answers
17 views

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 $T: Y \rightarrow X$ ...
2
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0answers
35 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
45 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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1answer
51 views

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
22 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 ...
2
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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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0answers
45 views

Solve the nth zero of a function. [closed]

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 ...
4
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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 [closed]

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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0answers
37 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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0answers
69 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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0answers
36 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
44 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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38 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 ...
5
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1answer
79 views
+50

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i'm not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, ...
1
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1answer
25 views

Corollary of the Birkhoff Kakutani Theorem: first countable topological vector spaces

http://planetmath.org/birkhoffkakutanitheorem A topological group $(G,*,e)$ is metrizable if and only if $G$ is Hausdorff and the identity $e$ of $G$ has a countable neighborhood basis. In ...
2
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0answers
41 views

Polar set of orthogonal matrices set is nuclear norm ball

Reltated problems: Show that the dual norm of spectral norm is Nuclear norm. Proof that nuclear norm is convex. The set of orthogonal matrices is defined as: $$\mathcal{O}(n) = \{X\in \...
2
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1answer
51 views

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$, $x\in X$, norm on $X$. Prove that with $(Ax)t = t^2x(a)$, ...
0
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1answer
34 views

For two positive operators on Hilbert space is it true that $A \ge B \implies \|A\| \ge \|B\|$?

$H$ is Hilbert space. $A$ and $B$ is positive linear operators from $H$ to $H$ i.e. $\forall x\in H\, (Ax,x),\,(Bx,x)\ge 0$. $A\ge B$ means that $A-B$ is positive. Does that means that $\|A\| \ge \|B\|...
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10 views

Convergence of operators and evaluation map for functions with values in a locally convex space

Let $E$ be a locally convex Hausdorff topological vector space, and $U$ a domain in $\mathbb{R}^n$. Suppose that we are given a continuous linear map $T: E \to C^{\infty}(U) \otimes E$. (The space $C^{...
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1answer
101 views
+250

States of a Group Ring

Let $G$ be a finite group and $\mathbb{C}G$ its group ring. Now taking the approach of orangeskid, consider the space $\mathbb{C}G$ as a Hilbert space with orthonormal basis $\delta^g$. $G$ acts on ...
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1answer
31 views

Continuous family of subalgebras in a C* algebra

Let A be a separable C* algebra. For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that: $A_t \cong \mathcal{O}_n$ (Cuntz algebra for fixed n). Generators of $A_t$ depend continuously ...
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55 views

Orthogonal of an Hilbert subspace and density

If $V$ is a subspace of an Hilbert space $H$, I know that the orthogonal of $V$, $V$$^o$, is ($V$closed)$^o$, even if $V$ is not closed. Does this mean that $V$ is always dense in $V$$^o$? Thanks!...
2
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1answer
39 views

Norm of convolution

Let $f,g: \mathbb{R}^n \to R$. Let $|| \cdot ||_T$ be a translation invariant norm on functions on $\mathbb{R}^n$. How can I prove that $||f*g||_T \leq ||f||_1 ||g||_T$ (where $f*g$ means convolution ...
4
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2answers
74 views

the uniform convergence of the sequence of functions

Let $f_1:[a,b]\rightarrow \mathbb{R}$ be a Riemann integrable function. Define the sequence of functions $f_n:[a,b] \rightarrow \mathbb{R}$ by $f_{n+1}(x)=\int_a^x f_n(t)dt,$ for each $n\ge 1$ and ...
1
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1answer
55 views

In a proof of the Riemann Lebesgue lemma

In a proof of the Riemann-Lebesgue lemma in Hunter's Applies Analysis, he first proves the statement in the Schwartz space and then uses a density argument: Here are my questions: What goes ...
2
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0answers
55 views

Frechet derivative of square root on positive elements in some $C^*$-algebra

Let $A$ - is some unital $C^*$ algebra, and $P$ is set of all strictly positive elements in $A$. We can define map $\sqrt{?} : P \to A$ which takes positive element and returns its (unique) strictly ...
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0answers
45 views

Differential $\sqrt{1+B(x,x)}$ map in $C^*$-algebra

Let $A$ is $C^*$-algebra and $P \subset A$ is subset of all elements $a \in A$ such that $a > 0$ (nonnegative) and $||a|| < \frac{1}{\sqrt{1-q}}$ (norm bounded) for some $0 < q < 1$. Let $...
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1answer
19 views

Show that $q(T)(x)=\sum_{n=1}^\infty q(\lambda_n) \langle x,e_n\rangle e_n$ coincide with $q(T)=\sum_{k=0}^n a_kT^k$

Let $Tx=\sum_{n=1}^\infty \lambda_n \langle x,e_n\rangle e_n$ be bounded where $\{\lambda_n\}_n$ are the complex eigenvalues and $\{e_n\}_n$ are an orthonormal basis of the separable space $H$. For ...
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1answer
75 views

How to write this as a function of only $z^2$?

$F(z)=\int_{{\mathbb R}^n}f(x)e^{2\pi x \cdot z-\pi x \cdot x-\frac{\pi}{2}z^2}dx$ Given $f\in L^2({\mathbb R}^n)$ is a radial function. $z\in{\mathbb C}^n$ and $z^2$ denotes $z \cdot z$ (dot ...
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33 views

Sufficient conditions for $f(T)$ to be compact and self adjoint whenever $T$ is compact and self adjoint

Let $Tx=\sum_{n=1}^\infty \lambda_n \langle x,e_n\rangle e_n$ be bounded where $\{\lambda_n\}_n$ are the complex eigenvalues and $\{e_n\}_n$ are an orthonormal basis of the separable space $H$. For ...
2
votes
1answer
50 views

Continuous function on the unit sphere [duplicate]

Let S$^2$ := $\lbrace$ x $\in$ $\mathbb{R}$$^3$ : $\Vert x\Vert$$_2$ $\rbrace$ $\subset$ ($\mathbb{R}$$^3$, $\Vert .\Vert$$_2$) and T: S$^2$ $\to$ ($\mathbb{R}$, $\vert x\vert$ ) a continuous function....
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0answers
24 views

Limit relevant to parametrised semi-group 2

Let $s\geq 1, \epsilon >0, T>0$ and $f \in \mathcal{C}([0,T], H^s(\mathbb{T}))$. Define the function $$g(t,x):= \int_0^t(Id-\exp\left(-i\tau\epsilon \Delta)\right)f(\tau,x)d\tau.$$ I want to ...
0
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0answers
25 views

Is the Lebesgue measure zero for the discontinuous set of a semicontinuous function?

[Q.] Is there a semicontinuous function, which has its discontinuous set with non-zero measure? Remark: Given a semicontinuous function, the set of all discontinuous points may be uncountable, for ...
0
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1answer
46 views

Convergence in $L^p$ and convergence almost everywhere

Why $f_n$ converges to $f$ in $L^p$ space implies that exists subsequence of $f_n$ converging to $f$ almost everywhere?
5
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1answer
59 views

Exponential of Operators

Let $H$ be an Hilbert Space $\exp(T)$ the exponential for an operator $T \in L(H)$. I know that $\exp(A)^{*} \exp(A)=\exp(A) \exp(A)^{*}=id$. Can I conclude that $A^{*}A=AA^{*}$? Cannot find an ...
0
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2answers
30 views

Any closed convex bounded set is weakly compact in a reflexive Banach space.

Let $X$ be a reflexive Banach space. Any closed convex bounded set is weakly compact. I know it is true. But, I can't find a reference. Anyone can help?
0
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1answer
23 views

Neighborhood of inclusion in space of Lipschitz maps is 1-1

Let $B \subset \mathbb{R}^n$ be the closed unit ball. Let $i(x) = x$ denote the inclusion map. Let $\|\cdot\|$ be any norm. Given $f:B\to \mathbb{R}^n$, define the sup norm $\|f\|_\infty:=\sup_{x \in ...
0
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0answers
26 views

Textbook/monograph for microlocal analysis

I want to grasp the theory of microlocal analysis and apply this theory to some PDEs in $R^n$. But most textbooks I found put much priority on manifolds. Sadly, I know little about them and don't ...
1
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0answers
29 views

Limit relevant to parametrised semi-group

Let $s\geq 1$, $T>0$, $\epsilon >0$ and $f\in\mathcal{C}^1(0,T,H^{s-1}(\mathbb{T}))\cap \mathcal{C}(0,T,H^{s}(\mathbb{T}))$. Consider the propagator $\exp\left[\displaystyle-\frac{it}{\epsilon}\...