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Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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11 views

Vector Space with Hamel Basis is not Separable when all basis elements are 2 apart

Consider a Hamel basis, $\{e_{\lambda}\}_{\lambda \in \Lambda}$, for an infinite dimensional linear vector space. I'm reading something that makes the following claim in passing: Note that if for ...
3
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0answers
11 views

A measure theoretic Lipschitz condition

Let $f$ be a measurable function satisfying following condition: \begin{equation*} \limsup_{r \to 0} \bigg\{ \frac{1}{\delta^N} \mathcal L^{2N} \Big( \Big\{ (x,y) \in \mathbb R^N \times \mathbb R^N : ...
1
vote
1answer
15 views

If $0 = \left(TH\right)^\perp = \ker T^*$, then $TH = H$.

If $H$ is a Hilbert space and $T$ is a bounded linear operator on $H$ with the following property $$0 = \left(TH\right)^\perp = \ker T^*$$ where $T^*$ is injective, then $TH = H$. Discussion I've ...
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0answers
16 views

Is $f \in L^2(M)$?

Let $$M=\lbrace rcos(\phi),rsin(\phi)\in \mathbb{R^2}:0<r<1,-\frac{\pi}{2}<\phi<\frac{\pi}{2}\rbrace$$ and $$ f(rcos(\phi),rsin(\phi))=rcos(\phi)$$ I have to show that $$ \int_M|f|^2d\mu &...
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0answers
15 views

Prove the orthogonal complement of a nonempty subset of a Hilbert space is closed.

Statetment Let $M \subset H$ be nonempty, and $H$ is a Hilbert space. Then $M^\perp$ is a subspace, and $M^\perp $ is closed. Proof To show that $M^\perp$ is a subspace we let $x,y \in M^\perp$, $a,...
0
votes
1answer
18 views

Square summable sequence in Hilbert space is summable?

Let $H$ be a Hilbert space. Is it true that if $h_n \in H$ is a sequence such that $\sum ||h_n||^2 < \infty$, then $\sum h_n$ exists? I believe that this is false, but I cannot construct a ...
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0answers
33 views

I don't understand the proof of Corollary 4.8.7 in the book of Ethier and Kurtz

I'm trying to understand the proof of Corollary 8.7 of Chapter 4 in the book Markov Processes: Characterization and Convergence by Stewart N. Ethier, Thomas G. Kurtz. Here is the theorem and its proof:...
2
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0answers
7 views

Examples of BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative

What are examples of two BV functions $u:\mathbb{R}^2 \to \mathbb{R}^2$ with singular derivative? More precisely, I'd like to see an example of a function $$u_1 \in BV(\mathbb R^2; \mathbb R^2)$$ ...
0
votes
1answer
29 views

Divergence in supremum norm implies divergence on whole segment

Let $X$ be the Banach space of the real valued continuous function in $[0,1]$ endowed with the supremum norm. Suppose $E \subset X$ is a finite dimensional subspace and take $\{f_n\}_{n \in \mathbb{N}}...
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0answers
24 views

A doubt on masa

Let $g$ in $L^{2}[0,1]$, let $f_{n}$ are in $L^{\infty}[0,1]$ such that $f_{n}{\rightarrow}^{\|\cdot\|_{2}} g$, from $f_{n}$ can we construct $g_{n}$ such that $g_{n}\rightarrow g$ in $\|\|_{2}$ norm ...
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0answers
19 views

Norm $\mathcal{C}^1$ multivariate functions

Given an open and bounded subset $\,\Omega\subset\mathbb{R}^n\,$ and the interval $\,[0,T]\,$ such that $\,T<\infty$, I'm trying to find a norm for $\,X=\mathcal{C}^1\left([0,T]\times\Omega\right)$ ...
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0answers
18 views

Faithful Representation of von Neumann Algebras

I have questions regarding part of E. Christopher Lance. Ergodic Theorems for Convex Sets and Operator Algebras, Page 205. I am looking for a good reference explaining how the faithful representation ...
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0answers
10 views

Is the generator of a uniformly continuous contraction semigroup contractive?

Let $E$ be a $\mathbb R$-Banach space and $A$ be the generator of a uniformly continuous contraction semigroup $(T(t))_{t\ge0}$ on $E$. Are we able to derive some bound on $\left\|A\right\|_{\mathfrak ...
1
vote
1answer
37 views

For continuously differentiable $f,$ is it true that the set $\{(x_0,x_1)\in (0,1)^2: |f(x_0)| + |f'(x_1)| \geq \epsilon\}$ not compact in $(0,1)^2?$

Notations: We denote $C_0^1(0,1)$ the collection of all real-valued continuously differentiable function $f$ on $(0,1)$ that vanish at boundary, that is, for any $\epsilon>0,$ the set $$\{x\in (0,...
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4answers
47 views

Solve $f(x)=c \times f(\frac{x}{2})$ for $c$

Given: Function $f(x)$ is infinitely differentiable equation (1) $f(x)=c \times f(\frac{x}{2})$ We have to find all $c$, for which the (1) has non-zero solutions Any hints on theorems to apply ...
0
votes
1answer
41 views

Find $b_n$ independent of $x$ satisfying $\sum_{n=0}^{\infty}\frac{b_n}{x^{n+1}}=0$

I am looking for non-trivial real coefficients $b_n$ independent of $x$ satisfying:$$\displaystyle\sum_{n=0}^{\infty}\frac{b_n}{x^{n+1}}=0$$ or prove that $b_n=0$ for every $n=0,1,2...$ Note that $...
1
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1answer
18 views

The relationship among different types of fundamental spaces.

I'm just looking to make sure my understanding of certain fundamental spaces are correct. Denote the set of all vector spaces by $V$, the set of all metric spaces by $M$, the set of all normed spaces ...
0
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0answers
31 views

Every Dunford-Pettis operator is compact

I was trying this problem. Let $X$ be a Banach space that does not contain a copy of $l^1$. Show that every Dunford-Pettis operator $T: X \rightarrow Y$, with $Y$ any Banach space, is compact. I ...
2
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1answer
34 views

About an equality of fractional Laplacian on a bounded domain

Let $0<s<1$. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. We know that $$\|(-\Delta)^{s/2}u\|_{L^2(\mathbb{R}^n)}^2=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+...
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0answers
31 views

For which $\gamma \in (0,1]$ does u belong to $C^{1,\gamma}(\overline K_{\beta}) $?

Let \begin{cases}\text{-$\Delta$u=f} \ \ in \ \ K_{\beta} \ \\\text{u=0 } \ \ in \ \ \partial K_{\beta} \\ \end{cases} $K_{\beta} \subset \mathbb{R^2}$ and $\beta \in(0,2\pi)$ define $$K_{\...
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0answers
21 views

Existence of a non bounded linear operator. [duplicate]

I want to prove using Hamel basis that for $X$, $Y$ normed spaces such that dim$(X)=\infty$ there exists a non bounded linear operator... but I do not know how to proceed. Any hint, please?
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0answers
22 views

Is my calculation correct?

For $\Omega=C_{\beta} \subset \mathbb{R^2}$ and $\beta \in(0,2\pi)$ define $$C_{\beta}=\left\{(r\cos(\phi),r\sin(\phi))\in\mathbb{R^2}|0<r<1,|\phi|<\frac{1}{2}\beta\right\}$$ and $\hspace{...
1
vote
2answers
61 views

$Lip_\alpha$ is not closed in $C[0,1]$

As the title says I am trying to show that $Lip_\alpha$ is not closed in $C[0,1]$. $Lip_\alpha$ is the class of functions on [0,1] that belong to $Lip_\alpha([0,1];K)$ where $f \in Lip_\alpha([0,1];K)$...
2
votes
1answer
65 views

Showing $h_n$ does not uniformly converge

$$f_n(x)=x(1+1/n) \text{ if } x \in \mathbb{R}$$ $$g_n(x) = \begin{cases} (1/n)& x = 0, \text{ or } x \in \mathbb{I} \\ b+1/n& \text{if } x \in \mathbb{Q} \text{ with } ...
0
votes
1answer
15 views

Solution of functional equation with sums and products

I'm not very familiar with functional equations,so I really need help. Is there some method to obtain functions $b_i, i=1,2$ from $$\displaystyle{\Big(a_1(u)b_1(v)+a_2(u)b_2(v)-\sqrt{a_1^2(u)+a_2^2(u)}...
0
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1answer
26 views

Discontinuous in every point function

Is there a discontinuous function at all points except $(0, 0)$, $f(x, y)$, for which for any $x_0, y_0$ functions $f(x_0, y)$ and $f(x, y_0)$ are non-constant?
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1answer
39 views

Have a question about Open Mapping Theorem in functional analysis homework

Let X and Y be Banach spaces. Prove that T ∈ B(X, Y ) is surjective if and only if range(T) is not a meager subset of Y. I have no clue..hope somebody help me.. thanks!
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0answers
26 views

Dual of an Algebra

I want to know that, as I know if I have two algebras and the morphism between them is an algebra morphism ... i am trying to figure out dual of that algebra morphism. For this I just need to take ...
0
votes
1answer
25 views

Linear operator image is not closed

Studying functional analysis, I have to prove that the image of the operator $S:\ell^1(\mathbb{N})\rightarrow \ell^1(\mathbb{N})$ given by $(S\xi)_n=\xi_n/n$ is not closed, even though $S$ is limited....
2
votes
1answer
15 views

Coarea-like formula for BV function (not its derivative)

Let $f \in BV(\Omega)$. The coarea formula states that $$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$ Do we also have that $$f = \int_{\mathbb R} \chi_{\{f >h\}} \, dh$$ holds?
0
votes
1answer
19 views

A convenient redefiniton of the Sobolev norm

I am dealing with the Sobolev space $W^{m,2}[0,1]$, i.e. the space of functions on $[0,1]$ with absolutely continuous $m-1$st derivative and square integrable $m$th derivative. I am using the norm $$|...
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votes
0answers
10 views

Subderivative of a product and of a quotient

Let $r$ be one subderivative of $f(x)$ at $x_0$ and $s$ be one subderivative of $g(x)$ at $x_0$. Does there exist a formula for a subderivative at $x_0$ of $h(x)=f(x)g(x)$ and of $i(x)=\frac{f(x)}{g(...
0
votes
1answer
26 views

Functional Analysis- invertible operators

Let ${c_n}$ $\in \ell^{\infty}$, and let $T_{c_n}$$\in$ B$(\ell^2)$, $\:$ be defined by $\\$ $T_{c_n}$ ({$x_n$}) = {$c_nx_n$}. $\\$ If inf {$\vert c_n \vert$: n $\in\mathbb{N}$} $\gt$ 0 , and $d_n$ =...
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0answers
36 views

Is this only definition?

If $u$ is twice differentiable and $\Omega$ a domain and $\Delta$ the Laplace operator then $$ -\Delta u \in C(\Omega) \Rightarrow u \in C^2(\Omega) $$ I think it is only definition . Thanks ...
1
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1answer
23 views

Total variation and Lipschitz continuity

Let $f:B_R(0) \subset \mathbb{R}^N \to \mathbb{R}$ be a $L$-Lipschitz continuous function. Is it true that the total variation $|Df|(B_R(0))$ is controlled by the Lipschitz constant $L$? How?
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votes
0answers
17 views

Characterization of summable families

Could somenone help me with to prove the following result? If $C=\{\lambda\in{\Lambda}:x_\lambda\neq{0}\}$ is countable and for every bijective mapping $\tau:\mathbb{N}\longrightarrow{C}$ $\sum_{n\...
1
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1answer
71 views

These two norms of $C^k[a,b]$ are equivalent.

In $C^k[a,b]$ we can define $$ \|f\|_* := \|f\|_\infty + \left\|f^{(k)}\right\|_\infty $$ and $$ \|f\|_{**} := \sum_{j=0}^k \left\|f^{(j)}\right\|_\infty$$ It is obvious that $\|f\|_*\leq \|f\|_{**}$...
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0answers
24 views

Technique to prove existence?

I would like to prove the existence of a T-periodic function $v$ in $H^1(R^N)$ s.t. $v=|v|e^{i\theta}$ for some T-periodic $\theta \in H^1(R^N)$ and s.t. one certain functional $I_c$ turns negative at ...
0
votes
1answer
13 views

outer measure on R some sort of continuity about Measure

$m^*(E)=q>0$,for any $c\in (0,q)$,there exist $E_0\subset E$,such that $m^*(E_0)=c$ $$m^*(E)=inf\{mG|E\subset G ,\text{G is open set}\}$$ I think if I can find a set A$\subset E$,and $m^*(E-A)=c$,...
0
votes
1answer
44 views

Eigenvalues and eigenfunctions of an integral operator

Let $T$ be an integral operator with kernel $K(x,y)=|x-y|$ on $L^2(-1,1)$. How can we find the eigenfunctions and eigenvalues of $T$? Even though I am not sure whether the following arguments are ...
1
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0answers
9 views

Point spectrum of an integral operator

Let we have $$Tu(x) = \cfrac{1}{x}\int_0^x u(y)dy$$ so that $u \in L^2(0,1)$. How can I show that $(0,2) \subset \sigma_p(T)$ and $T$ is not compact?
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0answers
15 views

$L^1$ estimate for difference between a certain function and its average

How can we estimate the difference $$\int_{\mathbb{R}^N} \left| \chi_{x_N \le F(x_1, \dots, x_{N-1})} - \int_{B_R(0)}\!\!\!\!\!\!\!\!\!\!\!\!\!\! - \ \ \ \chi_{x_N \le F(x_1, \dots, x_{N-1})} dx_{1,\...
1
vote
2answers
21 views

If $(e_n)_{n\geq 1}$ is total in $H$ and $\sum \| e_n - f_n \| < 1$, prove $(f_n)$ is total.

Sorry about the title. Not enough space for me. Proposition If $E = (e_n)$ and $F = (f_n)$ are orthonormal sequences in a real Hilbert space $H$ that satisfy $$\sum_{n=1}^\infty \|e_n - f_n\| < ...
4
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0answers
142 views
+100

Find function $f(x)$ satisfying $\int_{0}^{\infty} \frac{f(x)}{1+e^{nx}}dx=0$

I am looking for a non-trivial function $f(x)\in L_2(0,\infty)$ independent of the parameter $n$ (a natural number) satisfying the following integral equation: $$\displaystyle\int_{0}^{\infty} \frac{f(...
3
votes
1answer
21 views

Bounded linear operator $A$ s.t. $Ax=y$ has a least square solution for each $y$ iff the range of $A$ is closed

Let $A: H_1\to H_2$ be a bounded linear operator, where $H_1,H_2$ are Hilbert spaces. Prove that $Ax=y$ has a least squares solution for each $y\in H_2$ if and only if the range of $A$ is closed. I ...
1
vote
0answers
14 views

Relation between adjoint of trace operator and Dirac delta

Assume $u \in \mathscr{D}(\mathbb{R}^n)$ is a distribution of order $k$, with compact support on a smooth manifold $\Gamma \subset \mathbb{R}^n$. We know that we can write this distribution (Thm 2.3.5,...
1
vote
2answers
57 views
+50

For $\langle Tx, x \rangle \geq \|x\|^2$, prove a solution exists for $Tx = y$.

Edit I've posted this a couple other times, so I now plan on deleting those, and just using this one. Here's the original problem: $H$ is a real Hilbert space. Let $T: H\longrightarrow H$ be a ...
0
votes
1answer
45 views

Determining if the limit exists of the sequence

I am trying to determine whether the limit of the following sequence exists and if so, find the limit. $f$ is a positive continuous function of $[a,b]$. \begin{align*} \lim_{n\to\infty} \Big[\int_{a}...
0
votes
0answers
40 views

Functions as Vectors

Whenever I refer a book or video on how to represent a function as a vector, the source automatically assumes the function to be a polynomial $$a_0 + a_1 \alpha +a_2 \alpha^2 + ... + a_n \alpha^n $$ ...
2
votes
1answer
43 views

Closed graph and fixed points

I’m currently trying to understand the following Proposition from a paper i’m reading: Prop.: Let $X$ and $Y$ be two Hausdorff topological linear spaces. Let $H:X \times Y \rightarrow Y$ be a ...