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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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Spectrum of product of continuous linear operators

Note: Please do not give a solution; I would prefer guidance to help me complete the question myself. Thank you. Show that if $S,T\in\mathcal{L}(X)$, where $X$ is a Banach space over $\mathbb{C}$. ...
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16 views

Canonical Topology on Aut(V)

Suppose that $V$ is a Banach space and denote by $Aut(V)$ the group of linear automorphisms of $V$ onto itself. Is there a canonical topology on $Aut(V)$?
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20 views

Is there any video lectures online that follow Folland's Real Analysis?

are there any video lectures online that follow Folland's Real Analysis?
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31 views

How do I integrate this term?

Let $\Omega \subset \mathbb{R^n}$ , $u\in C^1(\Omega)$ , $\phi \in C^{\infty}_0(\Omega) $ a test function . $\int_{\Omega}\nabla |u| \phi dx=-\int_{\Omega}|u|\nabla\phi dx=-\int_{\{x\in\Omega|u>0\}...
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1answer
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Question in Volterra operator

Consider $V:L^2([0,1],\mathbb{C}) \rightarrow L^2([0,1],\mathbb{C})$ defined by $$(Vu)(x)=\int_0^xu(t)dt$$ (i) Calculate the adjoint $V^*$. $$V^*=\int_0^x \bar{u(t)}\ \ dt$$ (ii) Suppose $u \in C([...
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2answers
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Prove $ C^1[0,1]$ is not a Banach space with the sup-norm

Let $ C^1[0,1]$ be the normed space of continuously differentable functions on $[0,1],$ with $||x||=\max_{t\in [0,1]} |x(t)| $. Prove that $ (C^1[0,1],||.||)$ isn't a Banach space I think we need ...
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15 views

Intuition for commutant and bicommutant

Let $T$ be an operator on a $\Bbbk$-linear space $V$. The notions of commutant and bicommutant of $T$ seem to be important but I don't have any conceptual intuition for them. In hopes of mending this ...
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1answer
22 views

Graph norm of a closed operator

Let's say we have a separable Banach space ($B$, $\Vert \cdot \Vert$). $A: D(A) \to B$ is a closed operator defined on a linear subspace $D(A)$ of $B$. Define the graph norm on $D(A)$ by $$\Vert x \...
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20 views

Inner product with a norm

Prove that the $L^p$ norm, denoted $||.||_p$, $p \neq 2$ on $C([0,1])$ is not induced by inner product. Then, I have the hint Prove that for the functions $f(x)=1/2-x$ and $$g(x)=\left\{ \begin{array}...
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1answer
21 views

Operator with every complex number as an eigenvalue

The question : Let $V$ be the space of all functions $f : \mathbb Z \to \mathbb C$. Find an operator $T : V \to V$ such that every $\lambda \in \mathbb C$ is an eigenvalue i.e. for every $\lambda \...
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How to apply diffeomorphism theorem to obtain solvability of differential equation?

I've been reading about local and global diffeomorphism theorems recently and I started to wonder how shall I apply these theorems in order to solve nonlinear differential equations? Assume that we ...
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38 views

Prove that $T$ is not contraction and $T^2$ is contraction

$T$ is not contraction and $T^2$ is contraction"> For (a) $||(Tf)(t)-(Tg)(t)||= ||\int ^t_0 f(s)ds-\int ^t_0 g(s)ds||\\ =||\int ^t_0(f(s)-g(s))ds||\\=\sup_{0\le t\le 1}|\int ^t_0(f(s)-g(s))ds| \\ \...
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1answer
16 views

Define inner product on dual space.

V is a Hilbert space By Riesz Representation Theorem: $\forall f\in V^*\exists v$ s.t $f=l_v $ where $l_v(x)=<x,v>$ and $||l_v||=||v||$(Using this fact can check that norm of dual space ...
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Interpolation between log and polynomials using Riesz-Thorin

I consider two $L^1$ weighted spaces, with $m_1(x) = e + x, m_0 = \ln(e+x)$. It is known that the Riesz-Thorin interpolation theorem holds for $L^p$-weighted space. I have an operator $T: L^1(m_1) \to ...
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Is it true that: $u \in C^0(\bar U) \cap C^{\infty} (U) \Rightarrow u \in W^{k-1/p,p}(\partial U)$?

Let $U$ be an open, bounded subset of $\mathbb R^3$ with a $C^2-$boundary. For $k\ge 2$ and $1\le p<\infty$ is it true that: If $u \in C^0(\bar U) \cap C^{\infty} (U)$ then $u \in W^{k-1/p,...
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2answers
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If T is self-adjoint such that $||T||<1$ exist $(I+R)^2=I+T$

I have the following question if $T\in B(H)$ where T is self-adjoint and $||T||<1$and $H$ is Hilbert and separable, then $\exists R\in B(H)$ s.t $(I+R)^2=I+T$. Does anyone know where to start? All ...
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1answer
16 views

$L^2$-closure of polynomials in a complex variable on an open bounded set

The density of the polynomials $p(z,\bar z)$ in $L^2(\Omega)$, where $z\in\Omega\subset\mathbb{C}$ and $\Omega$ is an open bounded set, is the consequence of more or less standard facts in complex and ...
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2answers
51 views

Is this space equivalent to the James space?

The James space $J$ is a famous counter-example in functional analysis. It is an example of a Banach space that is isometrically isomorphic to its double dual, but is not reflexive. Define $$J = \...
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1answer
31 views

Constructing a counterexample for functions with L1 norm.

I am trying to disprove this statement for some functional analysis stuff, but I wasn't able to make much progress. Any advice is appreciated. Statement: Let $\{h_n\}$ be a series of functions ...
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0answers
23 views

Show that Roots of Orthogonal Polynomials are interlaced.

Does anyone have an easy proof to show how the roots of orthogonal polynomials $p_n$ and $p_{n-1}$ are interlaced. I have just shown that all roots of orthogonal polynomials lie in the interval $I$ ...
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0answers
13 views

direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps $$ 0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0. $$ We can take inductive limit (...
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1answer
24 views

Problem on the proof of Moreau-Yosida regularization Theorem

I do not understand some steps of the proof of the following theorem Theorem (Moreau-Yosida regularization). Let $X$ be a metric space, $f:X\longrightarrow\mathbb{R}\cup\{+\infty\}$ be a function ...
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1answer
22 views

Bilinear form on Hilbert Space is Lower Semicontinuous

I'm working on the following problem -- I just wanted to check the proof: Let $H$ be a Hilbert space and $a(u,v)$ be a nonnegative bilinear form on $H$. Prove $a(u,v)$ is lower semicontinuous with ...
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0answers
17 views

Finding a limit involving F(x) when certain conditions are given

I thought to determine the function first but5 since only one information is given and according to that f(x) has one root alpha and at that point, the derivative has to be zero. So I tried to assume ...
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0answers
30 views

Lp space is a Hom functor

Is there way to see $L^1$ a functor from the category with objects as metric measure spaces $(X,d,\mu)$ and morphisms Lipschitz maps to the category of Banach spaces (or something which has a ...
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0answers
12 views

What assumption makes the followin true? $ \lim_{h \to 0} \frac{1}{h}\int_{B_R(0)} 1 \wedge \left(f( x + h) - f( x) - ⨏_{B_h(x)} f dz \right) dx = 0 $

What assumptions on the function $f: \mathbb R^N \to \mathbb R^N$ make the following true? $$ \lim_{h \to 0} \ \ \ \frac{1}{h}\int_{B_R(0)} 1 \wedge \left(f( x + h) - f( x) - \int_{B_h(x)}\!\!\!\!\...
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0answers
15 views

what is a class of all linear similarities? [on hold]

Recently I have been reading an article, nad I couldn't find the definition of the class of all linear similarities ?.
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Sobolev space identification

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$. It is well known that $L^2(\Omega\times (0,T))$ can be identified with $L^2(0,T;L^2(\Omega))$. Now, let us consider $$H^{1}(0,T;H_{0}^{1}(\Omega))=\{...
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0answers
8 views

Approximation of extremal problem

Let $J$ be linear functional on some functional space $U$, $X_n \subset U$ -- some functional sets, $n = 1,2, ...$ such that $X_n \to X_0 \subset U $. And we have sequence of extremal problems: $\...
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1answer
36 views

For a function with mirror- and translational symmetry how can I find the domain where the function has a strict local minimum point?

Let $\ f:X \rightarrow {\rm I\!R}$ be a function and $X=\{(x,y)\in {\rm I\!R}^2\}$. The explicit expression of this function is unknown, but it can be assumed to be smooth and continuous. It is know ...
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0answers
35 views

Number of roots of $p_n-\lambda p_{n-1}$ where $p_n$ are orthogonal polynomials

Let $(p_n)_{n\in\Bbb N}$ be the orthonormal sequence of polynomials associated to a tempered weight $w$ on an interval $I$ (so that for example, $\deg(p_n)=n$). Show that $p_n-\lambda p_{n-1}$ has $n$ ...
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1answer
28 views

Show that $|T_{\epsilon}(f)-T_0(f)|\leq \frac{\epsilon}{2}||f||_E,\; \epsilon\in]0,1]$

I have the next exercice: For $\epsilon \in ]0,1]$, we have the function $$T_{\epsilon}(f)=\frac{1}{2\epsilon}\int_{-\epsilon}^{\epsilon}f(t)dt$$ and $T_0(f)=f(0).$ Show that for $\epsilon \in ]0,...
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0answers
20 views

Lebesgue's differentiation theorem for the function $f\to T_f$ from $L^p $ in $C^1(]-1,1[)$

We consider the space $E$ of the functions of class $C^1$ on $] -1, 1 [$ which are bounded as their derivative too. The norm of $E$ is $$||f||_E=\sup_{-1\leq t \leq 1}|f(t)|+\sup_{-1\leq t \leq 1}|f'...
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0answers
16 views

Eigenvalue estimate for Steklov type equation

Consider the Steklov eigenvalue Problem for the ball \begin{align} \Delta u &= 0 ~~~~~\text{in }~~ B_r \\ \partial_\nu u &= \lambda u~~~\text{in } \partial B_r. \end{align} It is well kown, ...
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8 views

Non separability implies $\sigma(W^*)$ not Borel

I am looking for a proof or counter example to the following: Let $W$ be an inseparable banach space and $W^*$ be its topological dual; then $\sigma(W^*)\neq B(W)$ for $B(W)$ is the Borel sigma ...
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1answer
22 views

Nonseparability of $l_2(…)$?

Let $X=l_2([0,1])$ be a space of sequences such that $x_n\in [0,1]$ for all $n\in \mathbb{N}$ with the standard norm of $l_2$. How to prove that $X$ is nonseparable?
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asymptotically left uniformly continuous and left uniformly continuous functions on semigroup S

Given a non-empty set $S$, we denote by $L^{∞}(S)$ the Banach space of bounded real-valued functions on $S$ with the supremum norm. Let $S$ be a semigroup. Then a subspace X of $L^{∞}(S)$ is left ...
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1answer
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Absurd $L^2\subset H^1$ containment from a compactness proof

Let $\Omega\subset\mathbb R^n$ be an open set, and recall $L^2$ and $H^1$ are the Hilbert spaces with inner products $(u,v)_{L^2}=\int_\Omega u\overline v\,dx$, and $(u,v)_{H^1}=\int_\Omega u\overline ...
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0answers
14 views

When Pettis/Dunford integral coincide with Bochner integral?

My question concers Pettis/Dunford and Bochner integral. Assume that $X$ is a reflexive and separable Banach space and $f\colon \Omega\to X$ is weakly measurable and $\int_{\Omega}\|f\| \,d\mu<\...
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1answer
18 views

Pointwise convergence implies convergence in the norm

Let $A$ and $B$ be normed spaces with norms $||\cdot||_A$ and $||\cdot||_B$ respectively, and let $\mathcal L(A;B)$ be the normed space of linear transformations from $A$ to $B$, with the norm $$||T||...
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1answer
28 views

$\ell^p(J)$ is complete (Banach)

I have to prove that the space $\ell^p(J)$ defined as the set of all functions $\psi: J\rightarrow \mathbb{F}$ s.t. $\psi$ is null except in a contable subset of $J$ and $||\psi||_p :=\bigg(\...
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0answers
23 views

$u(x)=c \cdot \lambda (\frac{x}{2})$ for $c$

Given: Function $u(x)$ is infinitely differentiable and $\lambda \in \Bbb{R}$ equation ${\bigcirc}\hspace{-4mm}{1}\space$: $u(x)= \lambda \cdot u(\frac{x}{2})$ find all $\lambda$, for which the ${\...
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0answers
10 views

Wave cone of the curl operator

How can one compute the wave cone $\Lambda_{\mathcal A}$, defined as \begin{equation*} \Lambda_{\mathcal A}:=\bigcup_{|\xi|=1} \ker \mathbb A^k(\xi) \qquad\textrm{with}\qquad \mathbb A^k(\xi)= (2\pi ...
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0answers
15 views

Precise Definition of Embedding Theorem in Sobolev Spaces

Rellich Embedding Theorem Let $\Omega$ be a bounded domain, then $H_{0}^{1}(\Omega)$ is compactly embedded in $L^{p}(\Omega)$ (denoted by $H_{0}^{1}(\Omega)\subset\subset L^{p}(\Omega)$) for $1\leq p &...
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1answer
33 views

Density in $H^1[0,1]$.

Why is $\{u \in C^2[0,1] | u'(0)=u'(1)=0 \}$ dense in $H^1[0,1]$?
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0answers
22 views

A functional is bounded on a bounded subset of Hilbert Space

Let $H$ be a Hilbert Space with the norm $||\,\cdot\,||$ and inner product $(\, ,\,)$. Define $I : H\to \mathbb{R}$ as a nonlinear functional on $H$. Definition 1(Differentiability of I) $I$ is ...
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1answer
27 views

Compactness of the set of finite Borel measures

Suppose $X$ is a compact subset of $\mathbb{R}^n$ for some $n \in \mathbb N$. Let $\mathcal M(X)$ denote the space of all finite Borel measures on $X$. Is $\mathcal M(X)$ compact under some commonly ...
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1answer
45 views

Question about bounds on $g(x) = \int_0^{\infty} \frac1{x+y} f(y)\,dy$

Let $$g(x) = \int_0^{\infty} \frac1{x+y} f(y)\,dy$$ where $x$ is (strictly) between $0$ and $\infty$. Prove that if $f$ is in $L_1(0, \infty$) then $g$ is not, and in general, if $f$ is in $L_\infty$, ...
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0answers
21 views

Diagonal representation of positive power of a compact positive operator.

Let $T$ be a positive, compact operator on a Hilbert space $\mathscr H$. Now $T$ is diagonalizable i.e. there is an orthonormal basis $\{e_i:i\in \mathscr I\}$ and a bounded set of complex numbers (...
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1answer
40 views

Fourier transform of $\varphi_m(u)=\int |x|^mu(x)dx$

I'm stuck with the following problem Let $\varphi_m \in \mathcal{S}'(\mathbb{R}^{n})$, $n \in \mathbb{N}$, $m\in \mathbb{C}$, $0 >\text{Re}(m)>-n$ the distribution defined by $$\varphi_m(u)...