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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8 views

A necessary and sufficient condition such that product of partial isometries is a partial isometry

I'm reading the paper P. Halmos, L. Wallen, Powers of Partial Isometries, Indiana Univ. Math. J. 19 No. 8 (1970), 657–663 (http://www.iumj.indiana.edu/docs/19054/19054.asp). And I got stuck on the ...
1
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0answers
18 views

Separation of closed convex sets in finite dimensional space

If $A, B \subset \mathbb R^n$ are closed, convex, and disjoint, is there a vector $a \in \mathbb R^n$ such that $a^t x < a^t y$ for all $x \in A$ and $y \in B$? I found many theorems requiring one ...
-1
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0answers
20 views

Creating a “nice” R^2->R function from partial subdomains

This question is probably poorly defined, mainly because I am still not entirely sure what I need, and I just need some ideas to start with. I have a R^2 domain, with a set of given curves ...
14
votes
1answer
884 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 ...
0
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0answers
18 views

Does a generator of an analytic semigroup with a compact resolvent has pairwise conjugate eigenvalues?

I am reviewing a paper in which the authors claim that their operator has eigenvalues $\lambda$ that are either real or pairwise conjugate (meaning that if $\lambda$ is an eigenvalue, then also the ...
1
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1answer
34 views

A Question on Normal Operators

Let $T$ be a compact operator on a Hilbert space $H$. I want to prove the following: If there exists Orthonormal Basis from the eigen vectors of $T$, then $T$ is normal operator.
2
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1answer
25 views

Involution and Gelfand Transform Properties

Let $\mathcal{B}$ be a commutative unital Banach algebra, and let for each $x\in\mathcal{B}$ $\hat{x}$ be the Gelfand transform. I assume that $\mathcal{B}$ has an involution *. I want to show that: ...
2
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0answers
33 views

a proof that $c_0$ is a Banach space

I'm currently reading the functional analysis lecture notes taught in MIT, and I came across a filling-in on the part of the reader. I wonder if I've filled in the details as expected by the author. I ...
0
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1answer
16 views

Dual of the Banach space of $k$-times continuously differentiable functions.

Let $C^k([0,1])$ denote the Banach space of $k$-times continuously differentiable functions $f:[0,1]\to \mathbb R$ with norm $$\|f\|_{C^k}:=\max_{i=0,\dots,k}\sup_{x\in [0,1]}|f^{i}(x)|.$$ I'm trying ...
4
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0answers
23 views

Lower bound on gap between consecutive eigenvalues on $L_2(\mathbb{R}^3)$

A similar version of this question was originally posted by me in the physics community, but it was suggested that I ask the mathematicians instead. So I have tried to strip off most of the physics ...
9
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0answers
245 views
+500

If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But let'...
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0answers
38 views

Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
2
votes
1answer
40 views

Is $C[a,b]$ isomorphic to $C([a,b]\cup[c,d])$

It is very simple to prove that $C[a,b]$ is isomorphic to subspace of $C([a,b]\cup[c,d])$ and vice versa $C([a,b]\cup[c,d])$ is isomorphic to subspace of $C[a,b]$. Is it true for $\bigl(C[a,b], C([a,...
4
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1answer
47 views

The completeness relation from QM in terms of inner products

I remember from QM that the completeness relation says $$ \sum_{n=1}^\infty |e_n\rangle \langle e_n | = I$$ so that $\langle x\mid y\rangle =\sum_{n=1}^\infty \langle x\mid e_n\rangle \langle e_n \...
21
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8answers
2k views

Why is uniqueness important for PDEs?

Every text on PDEs I come across will spend alot of time on showing the existence and uniqueness of solutions to a particular PDE. The importance of the existence of a solution to a PDE is obvious, ...
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0answers
6 views

The Set of Extremal Point Need not be closed

Exercise: Consider the Convex Hull $C$ of the points $(0,0,\pm 1)$ and the circle $\{(1+\cos\varphi,\sin\varphi,0): 0\leq \varphi \leq 2\pi\}$ in $\mathbb{R}^3$. Determine the extremal point of $C$. ...
0
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0answers
17 views

Radon probability measures on $X$: Determine the Weak$^{\ast}$ closure

Exercise: Equip the set $P(x)=\{\mu \in C_0(X,\mathbb{R})^{\ast}: \mu\geq 0, \Vert \mu\Vert=1\}$ with the weak$^{\ast}$-topology. There is a map $\delta:X\to P(X)$, $x\mapsto \delta_x$ given by: \...
3
votes
2answers
48 views

is this an counterexample for: $(C[a,b],\| \cdot \|_2)$ is complete?

our prof wanted to show that $(C[0,1],\| \cdot \|_2)$ is not complete. So he said $$f_k(x) = x^k$$ is a counterexample. I wonder if this is true. I tried to show that $f_k$ is cauchy sequence. But i ...
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2answers
2k views

Space of continuous functions with compact support dense in space of continuous functions vanishing at infinity

How can we prove that the space of continuous functions with compact support is dense in the space of continuous functions that vanish at infinity?
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1answer
59 views

Extreme points of the unit ball of the space $c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$

I want to prove that all "closed unit ball" of $$ c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\} $$ do not have any extreme point. Would you please help me? (Extreme ...
0
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0answers
8 views

$L^p([0,1])$ stricly convex [duplicate]

Exercise: For which $p\in [1,\infty]$ is $L^p([0,1])$ strictly convex? Solution: For strict convexity we have two equivalent definition: If $x\neq 0\neq y$ and $\Vert x+y\Vert=\Vert x\Vert+\Vert ...
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0answers
11 views

Fixed Point method and existence of entropy solution of a Nonlinear Parabolic PDE [on hold]

Could someone point me to literature that addresses this kind of problem? Thanks
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1answer
19 views

$C([0,1])$ strictly convex

My question is quite simple: Is the space $C([0,1])$ strictly convex? Where strict convexity is defined as: if $\Vert y+x \Vert=\Vert y\Vert+\Vert x\Vert$ then it implies that is exists an $\lambda&...
12
votes
1answer
474 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}\...
1
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1answer
394 views

Simple heat equation, solution regularity

I have a small problem with a regularity result for a simple parabolic heat equation: Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
2
votes
2answers
50 views

The space $BPV(0,1)$ is a separable metric space under certain metric.

This is exercise 2.42 from Leoni's book A First Course in Sobolev Spaces. The BPV is defined as the space contain the function $u$ such that $$ \text{Var}[u]:=\sup\left\{ \sum_{i=1}^n|u(x_i)-u(x_{i-...
5
votes
1answer
384 views

application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
1
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1answer
41 views

Compact operator problem: $I-T$ is onto, then $I-T$ is invertible?

I want to show the following: Let $T$ be a compact operator in a Hilbert space $H$. If $I-T$ is onto, then $I-T$ is invertible. Would you show me how to prove this argument? Or please tell me some ...
0
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1answer
37 views

A Spectrum of a compact operator in $\ell^p$

Let $\alpha_n \in \mathbb C$ and $\lim_{n\to\infty}\alpha_n = 0$. Let $T$ be a linear continuous operator from $\ell^p \to \ell^p (1\le p\le \infty)$ defined by $$ T((x_1, x_2, \ldots)) = (\alpha_1 ...
0
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0answers
17 views

uniform convergence of lagrange polynomials , exercise 12.16.15 dieudonne treatise vol 2

This is exercise 12.16.15 from Dieudonne's treatise on analysis volume 2 It attempts to find necessary and sufficient conditions on a sequence of control points in I = [0,1] for the lagarange ...
1
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1answer
31 views

What is the operator norm of $(c_1 x_1, c_2, x_2, \cdots)$ in $\ell^p$?

I'm wondering that the operator norm $\|T\|$ where $T:\ell^p \to \ell^p$, $1\le p\le \infty$ is $$ Tx = (c_1 x_1, c_2x_2, \cdots), $$ where $c_k \in \mathbb C$ such that $\lim_{k\to\infty} c_k = 0$. ...
0
votes
1answer
41 views

Compact operator with closed image

Let $K$ be a compact operator between two normed spaces. If $K(X)$ is closed, does this necessarily imply that $K(B)$ is closed? where $B$ is the closed unit ball?
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0answers
22 views

I don't see why $W^{1, 2}(\partial D)$ being compactly embedded in $L^2(\partial D)$ lets us show an operator is Fredholm of index zero.

Let $D$ be a bounded Lipschitz domain. Let $A$ be the single layer potential which maps $L^2(\partial D)$ into $W^{1, 2}(\partial D)$ boundedly. $A$ is given by: $$ A_D[\phi] = \int_{\partial D}G(x-y)...
0
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1answer
18 views

Exercise: The range of Compact Operators

Exercise: Suppose $K:X\to Y$ is compact operator. Show that $K(X)\subseteq Y$ is separable Assume $Y$ is a separable Banach space. Find a Banach space $Z$ and a compact operator $K:Z\to Y$ s.t. $K(...
2
votes
5answers
153 views

How to prove $ A^{\perp} $ is a closed linear subspace?

Suppose $ X $ is an inner product space and $ A\subseteq X $. I need to prove that $ A^{\perp} $ is a closed linear subspace of $ X $. Can anyone give me a idea?
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2answers
57 views

$C([0,1])$ is separable: Is my solution correct?

Claim: $C([0,1])$ is separable w.r.t. the supremum norm. My solution: We want a countable subset $M$ s.t. $\forall f \in C([0,1])$ it exists a $g_n\in M $ s.t. $\lim_{n\to\infty}\Vert g_n-f \Vert_{\...
1
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1answer
26 views

A problem concerning compact operators in $\ell^p$ [duplicate]

Let $\alpha_n \in \mathbb C$ and $\lim_{n\to\infty}\alpha_n = 0$. Let $T$ be a linear continuous operator from $\ell^p \to \ell^p (1\le p\le \infty)$ defined by $$ T((x_1, x_2, \ldots)) = (\alpha_1 ...
1
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1answer
35 views

A simple $\ell^1$ norm question

Let $B$ be a closed subspace of $\ell^1$ such that $$ B=\{\{x_n\}_{n=1}^\infty \in \ell^1 : \sum_{n=1}^\infty \frac{n}{n+1} x_n = 0\} . $$ I want to show that there is NO $x = (x_1, x_2, \cdots) \in B$...
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0answers
78 views
+100

How can we prove that $\langle\int_0^t\Phi_s{\rm d}W_s,x\rangle_H=\sum_{n\in\mathbb N}\int_0^t\langle\sqrt{λ_n}\Phi_se_n,x\rangle_H{\rm d}B_s^{(n)}$?

Let$^1$ $U$ and $H$ be separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U,H)$ be nonnegative and symmetric operator on $U$ with finite trace $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $...
0
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0answers
30 views

The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
6
votes
2answers
820 views

Image of closed unit ball under a compact operator

Let $X,Y$ be Banach spaces and $A\in\mathcal L(X,Y)$ . The task is to prove the following: $A$ is compact if and only if the image of the closed unit ball in $X$ is compact in $Y$. I have proven ...
1
vote
1answer
49 views

Space of $f(0)=f(1)$: Is Hilbert space?

Let $S$ be space consisting of collection of square integrable continuous functions $f:[0,1]\rightarrow\mathbb{R}$ with the constraint $f(0)=f(1)$. So $S$ is an inner product space with the inner ...
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2answers
31 views

$Im(K)\subset Y$ closed, infinite-dimensional: $K:X\to Y$ is not a compact operator

Proposition: If $K:X\to Y$ is a bounded linear operator between two Banach spaces $X$ and $Y$ such that $\operatorname{Im}(K)\subset Y$ is an infinite-dimensional closed subspace, then $K$ is NOT ...
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0answers
32 views

An Extreme Point of a closed ball of $\ell^\infty$ [duplicate]

I am trying to prove that all "closed unit ball" of $$ c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\} $$ do not have any extreme point. (Extreme Point) Let $X$ be a ...
0
votes
1answer
16 views

PDE argument about passage to limit in Bochner space and weak derivative

I have a function $u \in L^2(0,T;H^1_0)$ with $u' \in L^2(0,T;H^{-1})$, all on some smooth bounded domain $\Omega$. In addition I have a sequence $u_n$ such that $u_n \to u$ in $L^2(0,T;H^1_0)$ and ...
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0answers
14 views

Exercise involving sequence Palais-Smale and operator compact.

Let H be a Hilbert space and let K: H $\rightarrow{R}$ be $C^1$ and such that $\nabla K: H\rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H\rightarrow R$ defined by ...
0
votes
2answers
47 views

Bound on $f_h(t) := \frac 1h \int_t^{t+h}f$

Given $f \in L^2(0,T;L^2(\Omega))$ define $$f_h(t) = \frac 1h \int_t^{t+h}f(s)\;\mathrm{d}s$$ for $t \in (0,T-h)$ and $f_h(t) = 0$ for $t > T-h$. In this paper (http://dml.cz/bitstream/handle/...
3
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1answer
30 views

Algebra $A$ and its Gelfand spectrum

Let $A$ be the set of all function $f$ on $\mathbb{R}$ of the form $$ f(x)=d+\int\limits_{0}^{\infty}e^{ixt}k(t)dt,\qquad\quad x\in\mathbb{R}, $$ where $d\in\mathbb{C}$ and $k\in L_1([0,\infty])$. The ...
3
votes
1answer
62 views

A Question concerning $\ell^1$

Let $a_n >0$ and $\sum_{n=1}^\infty a_n < \infty$. Define $$ M=\{ \{x_n\}_{n=1}^\infty \in \ell^1 : \forall n, |x_n|\le a_n\} $$ Then I want to prove that $M$ is compact and $M$ cannot be ...
5
votes
5answers
8k views

Does there exist a function that is differentiable but not integrable? or integrable but not differentiable?

It has become very complicated to me to find out a function which is differentiable but not integrable or integrable but not differentiable.