Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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).

2
votes
0answers
78 views

Infinite dimensional Banach spaces and sequences

Let $X$ be an infinite dimensional Banach space. $a)$ Prove that there is a sequence $x=(x_n)_{n \geq 1}$ in $X$ such that $||x_n||=1$ for all $n$ satisfying \begin{equation*} \mathrm{dist}(x_{n+1},\...
2
votes
1answer
54 views

Help showing derivative is compact

I'm struggling with proving the following theorem: $\textit{Let }A:U\subset X\rightarrow Y\textit{ be a completely continuous operator from an open subset U of a normed}$ $\textit{space X into a ...
2
votes
0answers
21 views

Whittaker model equation

This is from Cogdell and Piateski-Shapiro's paper Derivative and L-functions for $GL_n$. Let $\lambda$ be a non-trivial $\psi$-Whittaker functional. The Whittaker model is defined as$W_v(g)=\lambda(\...
3
votes
2answers
39 views

Power Series of an Operator

I'm working through some functional analysis problems and am having trouble with the following. Let $f(z)=\sum_{n=0}^\infty a_n z^n$ be a power series with radius of convergence $R>0$. Let $A\in\...
-1
votes
1answer
43 views

For $x_n\to x$, is there homeomorphisms $f_n$ with $d(f_n(x_n), f_n(x))>\epsilon$ ? ($\forall n\in\mathbb{N}$)

Let $(X, d)$ be a compact metric space and $x_n\to x$ as $n\to \infty$. Let $\epsilon>0$ be given. Is there a sequence of homeomorphisms $f_n:X\to X$ such that $d(f_n(x_n), f_n(x))>\epsilon$ ...
1
vote
1answer
10 views

Linearity of functionals imply the inclusion of kernels

I understand most of the following claim's proof except statement I've highlighted below: Let $X$ be a linear space and $W$ be a subspace of $X^{\text{#}}$ (linear functionals - not necessarily ...
3
votes
1answer
25 views

find adjoint operator of an operator A

How to find adjoint operator of an operator A $$A \in B(C^1[0,1], C[0,1])$$ $$ (Ax)(t) = x'(t)?$$ In answer : for any functional $f_y$ originated by function $y \in BV_0[0,1]:A(f'_y) = g_z$, where ...
2
votes
1answer
60 views

Compactness of a specific set in weak topology

I have the following question: Let $E$ be a polish space (that is, a topological space, which is separable and metrizable, such that $E$ would be complete if equipped with this metric). Consider the ...
1
vote
1answer
22 views

How to conclude from a homogeneous function $f$ of degree $1$ differentiable at $0$ that all norms are never differentiable at $0$?

Assume the fact that a homogeneous function $f$ of degree 1 (i.e. a map from $E$ to $F$ such that $f(tx) = tf(x)$ for all $t > 0$) that is differentiable at $0$ is necessarily linear. How do we ...
0
votes
1answer
31 views

Open Mapping Theorem fails when space is not complete

I am working on the open mapping theorem, and I want to show that completeness is a necessary condition for it to work. I have shown that if $X$ is Banach, $Y$ is a normed space, then there exists a ...
2
votes
1answer
42 views

Scaled series of shifted continuous function

I encountered (a significantly more general version of) the following problem: Consider any sequence $(\alpha_k)_k\in \ell^1$ with $\alpha_k\neq 0$ for any $k$ and let $\psi:[0,\infty)\rightarrow\...
3
votes
1answer
42 views

$H$ Hilbert space, $T$ symmetric bounded linear, when is $H=R(T) \oplus N(T)$?

I just saw in an exercise that if I have a prehilbert space $H$ and $T$ a linear, bound and symmetric operator then $R(T)=N(T)^{\perp}$. Now I was asking myself whether $H=R(T) \oplus N(T)$. On wiki I ...
1
vote
0answers
99 views

3D-plots of complex functions

Commonly it's believed that one cannot fully visualize a complex function $f:\mathbb{C}\rightarrow \mathbb{C}$ because the full plot would have to be 4-dimensional. But is this true? And why would the ...
5
votes
1answer
42 views

There exists a linear operator with no proper invariant subspaces

Let $A$ be a bounded operator on a Hilbert space $H$ with two invariant subspaces $M$ and $N$ s.t. $N \subset M$, dim$(M \cap N^{\perp})> 1$, and have no invariant subspaces between $N$ and $M$. ...
5
votes
2answers
136 views

Compact subset of space of Continuous functions

Let $\mathscr{C}[0,1]$ denote the set of continuous functions with bounded supremum and let $K=\{f\in\mathscr{C}[0,1]|\int_0^1f(t)dt=1\}$. Then is $K$ compact in the space $\mathscr{C}[0,1]$? ...
0
votes
0answers
22 views

Question on embeddings of $W^{2,p}(\partial U)$ space

I sense that my question might be elementary but I'm quite puzzled about the following: Let $U \subset \mathbb R^3$ be an open, bounded and connected set with $C^2-$regular boundary $\partial U$. If $...
2
votes
1answer
14 views

Proof regarding quasi-solutions

I'm almost done proving a theorem regarding quasi-solutions, which I'm stating here: $Let\ A:X\rightarrow Y\ be\ a\ compact\ linear\ operator\ and\ let\ \rho>0.\ Then\ for\ each\ f\in Y\ there\ ...
1
vote
1answer
21 views

analysis of $T : f \to Tf$ with $[T(f)](x) = ie^{i\pi x}(\int_0^x e^{-i\pi t}f(t)dt - \int_x^1 e^{-i\pi t}f(t)dt)$

$\, f \in L^2([0,1],\mathbb{C})$ show that $T : f \to Tf, \, f \in L^2([0,1],\mathbb{C})$ is continuous, $[T(f)](x) = ie^{i\pi x}(\int_0^x e^{-i\pi t}f(t)dt - \int_x^1 e^{-i\pi t}f(t)dt)$ the ...
0
votes
1answer
19 views

About limit of variable x y

Suppose $a>0,b>0,r=\sqrt{x^2+y^2}$ 1)proof $x^ay^b=o(r^l)(r\rightarrow 0),0<l<a+b$ $2)\lim_{(x,y)\rightarrow (0,0)}\frac{x^ay^b}{r^{a+b}} $exist or not My attempt 1)$$\frac{x^ay^b}{r^...
1
vote
1answer
34 views

Prove that $(Au)(t)=\frac{d^{2}u(t)}{dt^{2}}$ is self-adjoint

Let $D(A)=\{ u \in L_2(0,T)| u, \frac{du}{dt}$ are absolutly continuous with $\frac{du}{dt} \in L_2(0,T)$, $u(0)=u(T)=0\}$ and, $(Au)(t)=\frac{d^{2}u}{dt^{2}}$ prove that $A$ is self-adjoint. Trial ...
1
vote
1answer
15 views

Brezis excercise. Does $h\in L_p(\Omega)$?

Given two functions $f,g\in L_p(\Omega)$ define $h(x)=max\{f(x),g(x)\}$ prove that $h\in L_p(\Omega)$. I know that $|h(x)|^p\geq |f(x)|^p$ and $|h(x)|^p\geq |g(x)|^p$ but this seems to go nowhere... ...
2
votes
1answer
34 views

show that $\ker(Id-T) = \ker(Id-T)^{\star}$

$H$ is a hilbert space and $T$ is a bounded linear operator on $H$, also $\|T\| \leq 1$ by calculating $\|Tx-x\|^2$ I have shown the following string of equivalences $$Tx = x \iff \langle\,Tx, x\...
1
vote
0answers
34 views

Brezis exercise, under what condition does $f$ belongs to $L_p(\mathbb{R^n})$

Brezis excercise 4.1: I am asked to show under what conditions does $f(x)=\{1+|x|^\alpha\}^{-1}\{1+|\log|x||^\beta\}^{-1}$ belong to $L_p(\mathbb{R}^n)$ where $\alpha,\beta>0$.. I know that for $f$...
2
votes
1answer
40 views

Show that the following function is continuous on $[0,1].$

Define $$\omega_f(\delta) =\sup \{|f(x)-f(y)|: (x,y)\in [0,1]^{2}\text{ and } |x-y|\leq \delta\}$$ where $f\in \mathcal{C}([0,1])$ and $\delta\geq 0.$ I have proven that for all $\delta_1,\delta_2\...
3
votes
2answers
61 views

What is the operator norm of $Tf(x) = x^2f(x)$?

Let $H = L^2([0,1],\mathbb{R})$ and $T : H \to H,\, Tf(x) = x^2f(x) $. $T$ is linear. $$\|Tf\|_{L^2([0,1],\mathbb{R})} = \sqrt{\int_0^1x^4f^2(x)dx} \leq\sqrt{\int_0^1f^2(x)dx} = \|f\|_{L^2([0,1],\...
0
votes
2answers
65 views

Boundedness of $f \mapsto f(0)$ w.r.t. different norms [closed]

Let $X = C[0, 1]$ be the space of all continuous real valued functions on $[0, 1]$. On $X$, we define two norms: For $f$ in $X,$ $$\|f\|_{\infty}:= \sup\{|f(t)| : t \in [0, 1]\} \quad \text{and} \...
0
votes
0answers
16 views

Interpolation theory and surjectivity

Assume $(A_0,A_1)$ $(B_0,B_1)$ are interpolation pairs, and assume $T$, is an operator $T$ which is continuous $A_0+A_1$ to $B_0+B_1$, from $A_0$ to $B_0$ (also surjective) and from $A_1$ to $B_1$ (...
2
votes
1answer
71 views

Inequality involving infinite norm.

Define $$\omega_f(\delta) = \sup \{|f(x)-f(y)|: (x,y)\in [0,1]^{2}\text{ and } |x-y|\leq \delta\}.$$ Let $F(f)(\delta) = \omega_f(\delta)$ for arbitrary $\delta\geq 0$ and $f\in \mathcal{C}([0,1]).$ ...
-2
votes
1answer
55 views

Given $T \in L(X,Y)$, show the equivalence between the existence of S:$S(T(x))=x$ and the fact that T is injective and $T(X)$ is complemented in $Y$. [on hold]

Given $X,Y$ Banach spaces and $T \in L(X,Y)$, show that the following senteces are equivalent: A) there exists $S \in L(Y,X)$ such that $S(T(x))=x$ for all $x \in X$. B) $T$ in injective and $T(X)$ ...
0
votes
0answers
34 views

Establish Archimedean property of a vector-lattice

I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem. I feel the statement below (or in fact weaker versions) should ...
1
vote
0answers
49 views

$\lim_{|h|\rightarrow 0} ||f(x+h)-f(x)||_p=0$ for $p =\infty$

For $f\in L^p(X)$ where $p\in [1,\infty)$. we have $$\lim_{|h|\rightarrow 0} ||f(x+h)-f(x)||_p=0$$ I use the fact that continuous function with compact support is dense in $L^p$ to prove the case for $...
2
votes
1answer
54 views

completeness and the open mapping theorem

Let $X,Y$ be normed vector spaces, I want to show that the open mapping theorem requires completeness of both spaces. So my question consists of two parts: $\textit{i)}$ Let $X$ be a Banach space and ...
0
votes
1answer
25 views

Limit of sequence of sequences

I got to thinking about sequences of Cauchy sequences. Here is a simple example. Let us define $b_n = (n,1,1,1,...)$ for $n\in\mathbb N$. So we have \begin{align} b_1&=(1,1,1,1,...)\\ b_2&=(2,...
4
votes
2answers
129 views

Funky function-composed within itself umpteen thousand times

$f(x)$ is a differentiable function satisfying the following conditions: $$ 0 < f(x) < 1 \quad \text{for all $x$ on the interval $0 \le x \le 1$.} \\ 0 < f'(x) < 1 \quad \text{for all $...
1
vote
1answer
34 views

Are the only eigenvalues of an operator $T^k$ the $\lambda_n^k$?

If I have an operator T which has eigenvalues $(\lambda_n)_{n \in \mathbb{N}}$, I understand that $\lambda_n^k$ are eigenvalues for $T^k$. But how to show that these are the only eigenvalues? Is this ...
0
votes
0answers
18 views

Is there an infinite dimensional version of Curvature?

Replacing coordinates $x^\mu$ with scalar functions of one variable $\phi(x)$. Then replacing the metric tensor $g^{\mu\nu}(x)$ with $G^{xy}[\phi]$. Then replacing $\frac{\partial}{\partial x^\mu}$ ...
0
votes
1answer
48 views

how to show the orthogonal projection? [closed]

Let be $(G,\circ )$ a finite group and $ \pi : G \rightarrow U(H) $ a group homomorphism . Set $V:= \{ x\in H: \pi(g)x=x, \forall g \in G \} $ Then the orthogonal projection on $V$ is given as ...
1
vote
0answers
28 views

A direct proof of the surjectivity of divergence operator

The main question that I want to solve is that: Let $\{X^k\}_{k=-\infty}^\infty$ be a sequence of (Banach) spaces which are subsets of distribution space $\mathcal D'(\Omega)$ where $\Omega\subset\...
0
votes
1answer
26 views

Boundedness and Strong convergence

$f_n\rightarrow f$ in $L^2(0,1)$, $\{ f,f_1,f_2,\ldots \}\subset H^1(0,1)$, $||f_n||_{H^1(0,1)}\leq M,\ \forall n\geq 1 $, Is is true that $f_n\rightharpoonup f$ in $H^1(0,1)$? If not, then what is a ...
-2
votes
1answer
29 views

Given a linear operator $T$ and a linear functional $\phi_n(x)=(T(x))(n)$, show that $T$ is continuous iff $\phi \in X*$

Given $X$ a Banach space, and $T:X \rightarrow l_p$ a linear operator, with $1 \leq p \leq \infty$, for all $n \in \mathbb{N}$ consider the linear functional $\phi_n:X \rightarrow \mathbb{K}$, defined ...
-1
votes
0answers
28 views

Prove that $\lim\limits_{n \to \infty} P(\Lambda_n | F_n) = 1_{\Lambda}.$

Let be $(F)_{n}$ filtration and $ A_{n} \in F_{n}$ for every $n \geq 0$. Let be $$ \Lambda_{n} = \bigcup_{m \geq n} A_m $$ and $$\Lambda = \bigcap_n A_n. $$ Prove that $\lim\limits_{n \to \infty} P(\...
-1
votes
1answer
35 views

How to bound uniformly this integral?

could someone help me with this question please? I need to prove that if $1<p<2<q\;$ and $f\in{L^p(\mathbb{R^2})}\cap{L^q(\mathbb{R^2})}$ then $$g(z)=\int_\mathbb{R^2}\dfrac{|f(y)|}{|z-y|}dy$...
4
votes
1answer
46 views

Why an unbounded operator is not-constructive relying on Hahn Banach theorem?

Let $T: \mathcal S(\mathbb R)\to L^2(\mathbb R)$ defined by $$Tf(x)=f'(x),$$ where $\mathcal S(\mathbb R)$ is the Schwarz space on $\mathbb R$. The question is : is there a continuous extension to $L^...
1
vote
1answer
18 views

Is the natural embedding from $X$ to $X^{**}$ a homeomorphism with respect to the weak topologies if $X$ is reflexive?

If $X$ is a reflexive Banach space, is it true that the natural embedding $\Lambda$ of $X$ into its double dual is a homeomorphism? Here we equip $X$ with the weak topology, and $X^{**}$ with the weak-...
0
votes
1answer
33 views

Prove $Tx=x$, for $x\in H$, if and only if $(Tx,x)=\|x\|^2$ and $\ker(I-T)=\ker(I-T^*)$

Let $H$ be a complex Hilbert space and $T:H\rightarrow H$ an operation such that $\|T\|\leq 1$. Show that $Tx=x$ if and only if $(Tx,x)=\|x\|^2$ $\ker(I-T)=\ker(I-T^*)$. My attempt 1. ...
0
votes
2answers
29 views

Why an operator $A:D(A)\to F$ is called “unbounded”?

Let $E,F$ Banach spaces. We says that a linear operator $O: E\to F$ is bounded if $$\|Ox\|_F\leq C\|x\|_E,$$ for a certain $C>0$. This definition makes sense because this is equivalent at $$\|O\|...
0
votes
1answer
20 views

Convergence in operator norm doesn't imply uniform convergence, examples?

I am learning about functional analysis and the relation between operator norm convergence and uniform convergence is not so clear to me. I think that uniform convergence imply convergence in operator ...
1
vote
1answer
54 views

$||T-T_n|| \rightarrow 0$ and $T_n$ are compact but $T$ is not a compact operator.

It is a result that if $||T-T_n|| \rightarrow 0$ in the norm operator an that the $T_n \in \mathcal{L}(X,Y)$ (were $Y$ is a Banach space) are compact operators, then $T$ is compact. I found from here ...
1
vote
0answers
25 views

Discuss the existence and uniqueness of the following operator equation

The question says Let $A:X \to Y$ where $X,Y$ are normed spaces. $(i)$ Discuss the uniqueness and existence of a solution of the operator equation $Ax=y$ where $x \in X$ and $y \in Y$ for different ...
1
vote
1answer
81 views

Extreme points of a closed unit ball

H is the closed unit ball of $\ell_1(\mathbb{N})$ Show that the set of extreme points of H is Ex(H)= $\{\lambda_{e_n} : \lambda \in \mathbb{C }, |\lambda| = 1, n ≥ 1 \},$ where $(e_n)_{n\geq 1}$ the ...