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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25 views

Energy functional of smooth map $S^2\to M$

Let $f:S^2\to M$ be a smooth map where $(M,g)$ is a Riemannian manifold. Typically the energy functional of $f$ is written as an integral over $S^2$, so it exists by compactness. However, in Ricci ...
3
votes
2answers
33 views

Analogy and connection between roots of unity and the solutions to $f^{(n)}(x)=f(x)$

In the number theory we have roots of unity, i.e. the $n$ solutions of: $$x^n=1,~~~n=1,2,3,\dots$$ $$x_1=1$$ $$x_2=(-1,1)$$ $$x_3=\left( 1, e^{\dfrac{\pi i}{3}}, e^{\dfrac{2 \pi i}{3}} \right)$$ $...
0
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0answers
9 views

Trace norm of an operator defined by its kernel

I search an upper bound of the trace norm of the operator $A$ on $L^2(\mathbb R;\mathbb C)$ defined by its kernel: $$K_A(x;y)=\int_{-a}^a \exp\Big(-\big|x-u\big|^2-\big|y-\frac{u}{b}\big|^2\Big) \, du ...
0
votes
0answers
12 views

Is most of the function whose domain is zero to positive infinity that can be drawn on a 2D graph nicely $C^{\infty}$ for all $x$?

I know that $x^{\frac{3}{2}}$ is not $C^{\infty}$ at $x = 0$. But I wonder for any function whose domain is zero to positive infinity that can be drawn on a 2D graph nicely is $C^{\infty}$. By nicely, ...
1
vote
0answers
23 views

Does boundedness in topological vector space imply boundedness in subspace

Let $X$ be a topological vector space, $Y$ be a subspace. If $M\subset Y$ is bounded in $X$ (i.e. for every open set $U$ of $X$, there exists $\alpha$,s.t. if $\left|\beta\right|>\left|\alpha\right|...
-1
votes
0answers
7 views

existence of entropy solution for a nonlinear degenerate PDE

I have come across this paper: UNIQUENESS OF THE ENTROPY SOLUTION OF A STRONGLY DEGENERATE PARABOLIC EQUATION, by Roberta Dal Passo. He tries to solve the equation $u_t=(\psi(u)\varphi(u))_x$,in R^...
0
votes
0answers
11 views

The polar topology of $X^{'}$ if $X$ is a barrelled space

Suppose $X$ is a barrelled space, $X^{'}$ is its dual. It is known that $X$ is the polar topology of all $\sigma(X^{'},X)$ bounded sets. Is it true that $A^{'}$ is the polar topology of all $\sigma(X,...
0
votes
1answer
25 views

Express in terms of the spectral decomposition of $A$ the set of $x, y$ for which an inequality is satisfied

I'm confused by this problem: Let $A \in \mathbb{C}^{n \times n}$ be diagonalizable with eigenvalues $0 \leq \lambda_1 \leq \cdots \leq \lambda_n$. Express in terms of the spectral decomposition ...
0
votes
1answer
27 views

duality in Banach spaces

Let $X$ be a Banach space. For every $x\in X,$ the non-empty duality set $\mathcal{J}(x)$ is defined as:$$\mathcal{J}(x):= \left\{j(x) \in X': \langle x, j(x)\rangle = \|x\|^{2} = \|j(x)\|^{2} \right\...
2
votes
1answer
21 views

$H^1(\mathbb{R}^3)$ vs $H^1_0(\mathbb{R}^3\!\setminus\!\{0\})$

I would like to understand whether the spaces $H^1(\mathbb{R}^3)$ and $H^1_0(\mathbb{R}^3\!\setminus\!\{0\})$ are the same or not. The first space is the standard Sobolev space, for which one also ...
1
vote
1answer
12 views

Differentiable Pointwise Limit Function of a Sequence of Differentiable Function.

We know that a point-wise limit function $f$ of a sequence of differentiable functions $(f_n)$ is not necessarily differentiable. The sequence of differentiable functions $$f_n(x) = \sqrt(x^2+\frac{1}{...
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votes
0answers
36 views

Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open $H:=L^2(\Omega,\mathbb R^d)$ $U$ be a separable $\mathbb R$-Hilbert space $Q:U\to H$ ...
0
votes
0answers
9 views

When $u \in L^q$ then $\partial u / \partial x_i \in W^{-1,q}$

Suppose $\Omega \subset \mathbb{R}^n$ is Lipshitz domain, $u \in L^q(\Omega)$. I am reading some article that assume as clear that $$\frac{\partial u}{\partial x_i} \in W^{-1,q}(\Omega).$$ Is that ...
0
votes
0answers
11 views

Can we write $\text{tr}[Q(x)^*\nabla^2u(x)Q(x)]$ for $u∈C^2(ℝ^d)$ and $Q:ℝ^d→\text{HS}(H,ℝ^d)$ in terms of a differential operator?

Let $H$ be a separable $\mathbb R$-Hilbert space $u\in C^2(\mathbb R^d)$ and $\nabla^2u(x)$ denote the Hessian of $u$ at $x\in\mathbb R^d$ $\operatorname{HS}(H,\mathbb R^d)$ denote the space of ...
0
votes
0answers
19 views

Thoughts about $\alpha-$Holder continuous functions

I was thinking about the relationship of the $\alpha-$Holder continuous functions and their derivatives. The question is: If $f\in C^{k,\alpha}$, whats the better conditions to $f^{(k+1)}$? For ...
0
votes
1answer
21 views

Is there a unique projection map in this case?

Let $X$ be a Banach space over $\mathbb{C}$. Let $A,B$ be closed subspaces of $X$ such that $X=A\oplus B$. Assume that $||a+b||=||a||+||b||$ for each $(a,b)\in A\times B$. Then, does there exist a ...
6
votes
1answer
45 views

Do eigenvalues of a linear transformation over an infinite dimensional vector space appear in conjugate pairs?

While attempting to answer a question here (namely, the finite dimensional case of the title question: Prove that if $\lambda$ is an eigenvalue of $T$, a linear transformation whose matrix ...
0
votes
0answers
19 views
+50

$ι:U→V$ is an embedding, $Q:=ιι^*$, $L∈𝓛(ℝ^d)$, $Φ∈\text{HS}(U,ℝ^d)$ $⇒$ $\text{tr}LΦ\sqrt Q(Φ\sqrt Q)^*$ doesn't depend on $ι$

Let$^1$ $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota\in\operatorname{HS}(U,V)$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $u:\mathbb R^d\to\mathbb R$ be twice Fréchet ...
11
votes
1answer
132 views

Is there a probability measure on $[0,1]$ with no subsets with measure $\frac{1}{2}$?

I have a decidedly weird question. Does there exist a probability measure $(\mu, \mathcal{F})$ on $[0,1]$ such that 1) $\mu(x) = 0$ for every $x \in [0,1]$ 2) For every $r \in [0,1] \setminus \...
0
votes
0answers
11 views

A matrix decomposition problem for row/column element order

Indeed, I don't know how to classify this problem, but I try to use matrix to describe it. The problem is that there exists a function $f(x, y)$ and its exact form remains unknown. But I have some ...
0
votes
1answer
32 views

The composition of a dissipative operator and a positive opeartor is dissipative?

Let the real Hilbert space $H^1(\Omega)$ endowed with its usual inner product, denoted by $\langle ., . \rangle$ and let $A : H^1(\Omega) \rightarrow H^1(\Omega)$ be a dissipative ...
0
votes
1answer
27 views

Closure of a set in the weak topology

Let $X$ be a Banach space, $S$ a subset of $X$. What is the closure of $S$ with respect to the weak topology?
0
votes
0answers
35 views

Does Taylor series around point zero (maclaurin series) always exist for differentiable function?

For every differentiable function $f(x)$, is $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} \, (x)^{n}$ always true and can be written? The only thing we have to be concerned is just whether ...
-1
votes
0answers
12 views

Domain Definition for Gradient Operator

I want to define to gradient operator $\nabla f$ on an $n-$variable function $f\left(x_1,\cdots,x_n\right)$. For this purpose I want to well define the spaces that the domain and the range of the ...
0
votes
1answer
32 views

Difference between little o and big O notation in taylor expansion

I know I can say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+o(\Delta t) $$ But can I say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+O((\Delta t)^2) $$ In both cases, do I have to add that $\Delta t \...
0
votes
1answer
40 views

Is there a name for the “scalar product” in Banach spaces?

What would be the best term for the same object i.e $(x,y)$. I guess its some kind of bilinear form, but in general it does not obey the parallellogram law hence its not a scalar- or innerproduct. ...
1
vote
1answer
63 views

“Of the order of” notation

I have a function where all terms have the same coefficient $x^3$ in it: For example $f(x) = ax^3 - bx^3$ Can I say that in big $O$ notation: $f(x) = O(x^3)$ $f(x) = O(x^3)$ as $x \rightarrow 0$ ...
1
vote
1answer
24 views

Show space of C1 functions on (0,1) is a Banach lattice

I'm working on the beginning of a book on Banach lattices, and it wants me to show that $C^1(0,1)$ is a Banach Lattice, with the norm $\|f\|=\|f'\|_\infty+|f(0)|$ and the order $f\le g$ if $f(0)\le f(...
1
vote
1answer
33 views

Differentiability in fractional Zygmund spaces

I'm trying to understand why for $s$ not an integer, if $$S_0u\in L^\infty\text{ and }\sup_{j\geq0}2^{js}\|\Delta_ju\|_{L^\infty}<\infty,$$ then $u\in C^s$. Here I write $\Delta_j$ for the ...
1
vote
1answer
22 views

How do you define the inverse of an (exponential Lie) operator?

I know this is a fairly general question, but I would like to know anything I can about obtaining the inverse of an exponential of a lie operator. More specifically, I want to know how one can ...
1
vote
1answer
36 views

How tight is this trace inequality?

I would like to know how tight the following trace inequalities are for real symmetric $A$ and real symmetric $B \succeq 0$ $$\mbox{trace} (AB) \leq \lambda_{\max} (A) \cdot \mbox{trace} (B) $$ or ...
1
vote
1answer
16 views

Addition of $X^*$ is continuous w.r.t. Weak Topology $\sigma(X^*, X)$

I showed that the addition on $X^*$ is continuous w.r.t. Weak Topology $\sigma(X^*, X)$. Since I'm a newbie in this field, would you please check my proof and answer that my proof is correct or not? ...
2
votes
0answers
18 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 ...
2
votes
1answer
29 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 ...
0
votes
0answers
27 views

Generator of an analytic semigroup with a compact resolvent --> 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 ...
2
votes
1answer
38 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: ...
4
votes
0answers
31 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 ...
1
vote
0answers
42 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
48 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,...
0
votes
1answer
19 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 ...
0
votes
0answers
10 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 $C$. ...
1
vote
1answer
37 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.
0
votes
0answers
18 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: \...
0
votes
0answers
9 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 ...
-1
votes
0answers
29 views

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

Could someone point me to literature that addresses this kind of problem? Actually I want to prove the existence of entropy solution for a nonlinear degenerate PDE for an equation that cannot solve ...
0
votes
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&...
1
vote
1answer
43 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
votes
1answer
44 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
votes
0answers
21 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
vote
1answer
33 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 $\lim_{k\to\infty} c_k = 0$. ...