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

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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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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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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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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78 views

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 ...
3
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1answer
36 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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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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421 views

Is the derivative of a function square integrable if the function itself is square integrable in an interval?

If a function $\Phi(x)$ is square integrable in an interval $[-L, L]$, is the function $\Phi'(x)$, the derivative of $\Phi$ with respect to $x$, necessarily square integrable in the same range? In ...
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21 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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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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28 views
+50

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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15 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
18 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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+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, ...
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0answers
8 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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1answer
21 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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14 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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2answers
39 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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1answer
32 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
33 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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1answer
26 views

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
74 views

Exercise about spectrum of selfadjoint operator.

I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space. The problem is the following: Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a ...
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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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66 views

Prove that there is a dense subset of X on which $f$ is continuous.

Let ${\left(f_n\right)}$ be sequence of continuous function on a complete metric space $X$ which converges point-wise to a function $f$ then prove that there is a dense subset of $X$ on which $f$ is ...
4
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1answer
92 views
+50

Bessel-like inequality

Let $\{e_n\}$ be an orthonormal sequence in an inner product space E. Then I'm trying to show the following inequality: $$\sum_1^\infty| \langle x, e_n \rangle \langle y, e_n \rangle | \leq ||x||\...
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13 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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63 views

Finding all possible values of a Function

Let a function be defined as $f:N\to N$ and $x-f(x)= 19\left[\dfrac{x}{19}\right] - 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$ and $1900<f(1990)<2000$. Find all values of $f(1990)$. $...
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1answer
49 views

Proving continuity of a operator $T\colon E \to E'$ [duplicate]

Let be $E$ a Banach space over real numbers and $T\colon E \to E'$ linear such $T(x)(x)\geq 0$ for all $x\in E$, prove T is continuous. If $x_n\to x$ and $T(x_n)\to \phi\in E'$ then $T(x_n)(y)\to\...
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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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108 views

Partial isometry and projection

The following is a Theorem of Murphy's C*-algebras and operator theory: Let $H_1, H_2$ be Hilbert spaces and $u\in B(H_1,H_2)$. If $u^*u$ is a projection, then $uu^*u=u$. To show it, for $\xi\in H_1$...
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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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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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1answer
30 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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2answers
35 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^{\...
2
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0answers
31 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 ...
2
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0answers
35 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 \...
4
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1answer
49 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-...
14
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1answer
864 views

Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
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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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1answer
83 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F \...
3
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1answer
75 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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43 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
20 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
21 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 ...
12
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1answer
461 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: $$\frac{d}{dt}\...
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42 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)? ...
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0answers
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 ...
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1answer
23 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 ...