Stack Exchange Network

Stack Exchange network consists of 175 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
1answer
26 views

To Prove $T $ is a self map and $T$ have no fixed points

Let $K=\{x=(x(n))\in c_0:0\le x(n)\le 1$ for all $n\in \mathbb{N}\}$. Define $T:K\to c_0$ by $T(x)=(1,x(1),x(2),x(3),...).$ Prove : (a) $T$ is a self map on $K$ and $||Tx-Ty||_\infty=||x-y||_\infty $ ...
2
votes
0answers
20 views

$ W_{0}^{2}(\Omega)=\{ f\in W_{0}^{1}(\Omega):\Delta f\in L^{2}(\Omega)\}? $

Let $\Omega\subset\mathbb{R}^{n}$ be an open bounded domain. Let $W^{2}\left(\Omega\right)$ be the usual Sobolev space $$ W^{2}\left(\Omega\right)=\left\{ f\in L^{2}\left(\Omega\right):f,\partial_{i}...
2
votes
1answer
23 views

Confusions in Evans book regarding weak derivatives in Banach spaces

I am studying PDE using Evans' book and I have two main confusions (probably stupid questions to experts) regarding weak derivatives in Banach spaces. First confusion: $\def\u{\mathbf u}$ $\def\v{\...
0
votes
1answer
20 views

Show that each $y\in Y_1$ has a unique $x_y\in X.$

Setting: Let $X,Y$ be compact Hausdorff spaces and $E,F$ be any Banach spaces. Let $C(X,E)$ be the collection of all $E$-valued continuous functions on $X.$ $C(Y,F)$ is defined similarly. Endow sup-...
0
votes
1answer
18 views

The existence of boundary defining function

Let $\Omega\subset\mathbb{R}^n$ be an open set. For each $x\in\partial\Omega$ there exist a ball $B(x,r)$ and a function $f\in C^1(B(x,r),\mathbb{R})$ with the property $\nabla f\neq 0$ on $B(x,r)\cap\...
0
votes
0answers
25 views

Convergence of scalar product in a hilbert space

Background: From this paper I'm trying to understand why the OPE in conformal field theory has a finite radius of convergence. The authors make the claim that the scalar product of two states ...
7
votes
1answer
69 views

Heat semigroup norm between fractional Sobolev and $L^p$ spaces

What is the actual inequality that holds for the heat semigroup between fractional Sobolev space $W^{2\alpha,p}$ and classical Lebesgue space $L^q$? I am trying to derive an inequality $$ \lvert\...
3
votes
2answers
2k views

Is this a Projection operator on Hilbert space?

Let $T$ be a bounded operator on a Hilbert space with the property that $T^*(T-I)= 0$. I'd like to show that $T$ is an orthogonal projection. I'm not really sure how to show that an operator is an ...
1
vote
2answers
34 views

Show that $f_n$ converges in $L^1$ norm

Suppose that $f_n: X \to C$ are a dominated sequence of measurable functions, and let $f:X \to C$ be another measurable function. Show that $f_n$ converges in $L^1$ norm to $f$ if and only if $f_n$ ...
1
vote
0answers
17 views

Example of Drazin invertible operator that is not invertible

Let $ X $ be a Banach space. A bounded operator $ T $ in $ X $ is said to be Drazin invertible if there exists $ k \in \mathbb{N} $ and a bounded operator $ T^D $ in $ X $ such that $ TT^D = T^DT $...
1
vote
0answers
31 views

Calculating infinite sum using Parseval's theorem

For$\alpha \in \mathbb{R} \backslash \mathbb{Z}$, consider the fnunction $[0,2\pi] \to \mathbb{C} : x \mapsto \frac{\pi}{\sin \pi \alpha} e^{i(\pi - x)\alpha}$, and prove that $\sum_{n=-\infty}^\infty ...
0
votes
1answer
36 views

Numerical Approximation Solution to Exponential Equation

I have a question about finding an approximate value $x$ for the following expression: $$\frac{(e^{x\alpha_{1}})^2 + (e^{x\alpha_{2}})^2 + \ldots + (e^{x\alpha_{n}})^2}{\displaystyle \left(\sum_{i = ...
1
vote
2answers
45 views

A questions on Functional Analysis

Let $M=\{f\in C_{\mathbb{R}}([0,1]): f(0)=0\le f(t)\le f(1)=1,$ for $t\in [0,1]\}$ where $C_{\mathbb{R}}([0,1])=\{f:[0,1]\to \mathbb{R}:f$ is continuous on $[0,1]\}$ is Banach space with norm $||f||_\...
9
votes
3answers
287 views

Motivation for test function topologies

I'm a phisicist, who started looking just a little bit into distribution theory, so I can claim to know what I'm doing when throwing about dirac-deltas. Hence I only know two test function spaces: $\...
0
votes
0answers
21 views

Approximating $\max\frac{x_\tau}{x}$

I have the following delay system: $$x'(t) = g(t,\tau,x)$$ Given that $g(\cdot)$ is smooth and bounded, $x(t)$ is bounded in a positive region. What are some possible ways to obtain an upper bound on $...
0
votes
0answers
7 views

Is it true that $W=(W\cap JW)\oplus (W\cap JW^\perp)$?

Statement : Let $(V,(\cdot,\cdot))$ be a real Hilbert space. Let $J\in \rm{Aut}(V)$ be a complex structure on $V$, i.e. $J^2=-\rm{id}_V$. Suppose that $J$ is compatible with the inner product so that ...
3
votes
1answer
33 views

If series is not uniformly convergent, can we still integrate term by term?

We know that if $\sum a_nx^n$ converges uniformly, then we can integrate term by term. So this is just a sufficient condition, right? Does there exists a series not converging uniformly and still we ...
1
vote
1answer
47 views

About the locally convex topology

I know that if a locally convex space Hausdorff $(X,S)$ is first numerable then for the $\hat{0}\in X$ exists a countable local base $\{V_n, n \in \mathbb{N}\}$ and to each $V_n$ corresponds a ...
1
vote
0answers
26 views

Proof that $\frac{e^{st}}{2\pi i}$ is an orthogonal basis.

I was studying the Linear Algebra perspective about the Laplace Transform. We know that the Laplace Transform is given by: $$ F(s) = \int_{0}^{\infty}f(t)e^{-st}dt $$ Where $e^{-st}$ is the integral ...
1
vote
1answer
20 views

weak convergence and compactness

please how can I prove that if a sequence $u_n \to u$ in $L^{\infty}(\mathbb{R}^+; H^1(\Omega)) $ weak * and $\partial_t u_n \to \partial_t u$ in $L^2(0,T, L^2(\Omega)) $ weak for all $T>0$ ...
0
votes
0answers
16 views

What's the difference between a singular function and a singular continuous function?

I am a physicist so I am trying to make sense of definitions. As far I know, a singular function on $[a,b]$ is defined as: $f$ is continuous on $[a, b]$. the derivative $f′(x)$ exists and is zero ...
1
vote
1answer
24 views

False statement that all norms on the direct sum of normed spaces are equivalent.

I am currently working on the exercises in Conway's A Course in Functional Analysis and I think the following problem is not true. Here $\oplus_p X_k = \{(x_1, ..., x_n) \in \oplus_{k=1}^n X_k: (\...
0
votes
1answer
29 views

what is $W^{-1,2}$?

I have been reading about the weak Poincare lemma in the book "Linear and Nonlinear Functional Analysis with Applications" by P.G. Ciarlet. Let $\Omega$ be a simply connected domain in $\mathbb R^n$...
0
votes
0answers
23 views

Exemple of an homogeneous function

Is there an example of a smooth homogenous function $f$ of degree $r\ge3$ (which is not a polynomial) defined in $\mathbb{R}^{d}$ satisfying the following assumption $\partial_x^{\alpha} f(0,x_2,x_3,....
2
votes
1answer
682 views

Cartesian product of reflexive spaces is reflexive

Given $(E,\|\|_E),(F,\|\|_F)$ reflexive normed vector spaces. I have to prove that also $(E\times F,\|\|_{E\times F})$ is reflexive where $\|\|_{E\times F}$ is the product norm. What I know is that $(...
1
vote
0answers
42 views

Name of a set in $\mathbb{R}^{d}$

What is the name of the set \begin{align*} \mathrm{supp}\;u_j\subset\left\lbrace q\in\mathbb{R}^{d}, \frac{1}{2}\le |q|\le 3\right\rbrace. \end{align*} For $d=2$ we say an annulus. And for $d\ge 3$ ...
1
vote
0answers
17 views

Linearized operator: Fredholm operator? Jordan form?

Suppose we are given a reaction diffusion equation $$ u_t=u_{xx}+f(u) $$ and search for travelling wave solutions $u(x,t)=U(x-ct)=U(\xi)$ then plugging this into the equation we get $$ U_{\xi\xi}+cU_\...
0
votes
2answers
35 views

If A is a closed linear subspace of a vector space X (which may not be Hilbert) is it true that $A^{\perp} = 0 \Leftrightarrow A = X$?

If A is a closed linear subspace of a vector space X (which may not be Hilbert) is it true that $A^{\perp} = 0 \Leftrightarrow A = X$? $A^{\perp} = \{x \in X|<x,a>=0, \forall a \in A\}$. ...
0
votes
0answers
17 views

Conditions about Q is a operator on $C(M^2, [0, 1])$ the uniformly continuous functions space

Let $C(M^2, [0, 1])$ be the space of uniformly continuous functions, i.e. $f \in C(M^2, [0, 1])$ iff $f: M \times M \rightarrow [0,1]$ is a uniformly continuous function and $Q$ a "operator" such that ...
1
vote
1answer
37 views

$\forall\varepsilon > 0,\exists\ a >0 : |f(x)|\,\le\, a\|f\|_2 + \varepsilon\|f'\|_2$ for $f\in H^1(0,1)$

I know that function evaluation in $H^1(0,1)$ is continuous (see, e.g., Is the Delta distribution a continuous functional on $H^1(\mathbb R)$). So, $\delta_x : H^1(0,1)\to\mathbb C$ is a continuous ...
0
votes
0answers
26 views

Showing $||f - S_{n}(f)||_{2}^{2}$ $=$ $||f||_{2}^{2}$ $-$ $||S_{n}(f)||_{2}^{2}$

I am learning about Fourier Series in Carother's Real Analysis. We have just learned the Fourier Partial Sum $S_{n}(f)$ is the closest function to $f$ in the set of trigonometric polynomials with at ...
-1
votes
0answers
16 views

Use structure preserving maps to characterize “structures induced by another structure”?

For any set $X$, let $N_X$ be the subcategory of normed vector spaces with underlying set $X$. Similarly, let $M_X$ be the subcategory of metric spaces with underlying set $X$. Then (if I understand ...
3
votes
2answers
19 views

Does a closed set with a non-empty “half-core” necessarily have an interior?

Background: Suppose $X$ is a Banach space. Denote by $[x, y]$ the line segment between $x$ and $y$. Given $C \subseteq X$, we define $$\operatorname{core} C = \{c \in C : \forall x \in X, \exists\...
0
votes
1answer
30 views

Are there examples of sequences $(x_n)$ such that $(x_n) \in l^p(\mathbb{N})$ but $(x_n) \not\in l^p(\mathbb{Z})$?

I'm curious about whether/how a bi-directional sequence can have stricter conditions for convergence than a sequence over $\mathbb{N}$. I assume that this is possible, but haven't been able to ...
0
votes
1answer
16 views

Boundedness of a function implies coercivity of certain functional

I am reading a paper, which states Let $\Omega=(x_0,y_0)$ be an open interval in $\mathbb{R}$ and let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Borel function. Consider the functional $$ F_\Omega: ...
1
vote
1answer
44 views

How can I argue that $L\{(x_1,x_2,…,x_n,…)\} = \{x_1, x_2/2, x_3/3 … x_n/n …\}$, $L: \ell_2 \rightarrow \ell_2$ is bounded operator?

How can I argue that $L\{(x_1,x_2,...,x_n,...)\} = \{x_1, x_2/2, x_3/3 ... x_n/n ...\}$, $L: \ell_2 \rightarrow \ell_2$ is bounded operator? I think that it's intuitively since $\ell_2$ is sequences ...
-1
votes
1answer
49 views

Can someone help? need to find the a and b for this function. Anything helps!!

enter image description here Function is $f(x)= ax^4-25x^3+2x^2+25x+b $ it passes through (7,E) E= 1296 and there is a zero of -1
0
votes
1answer
21 views

Calculus of Variations by Charles Fox: Question on Statement in Section 2.4

Fox states in Section 2.4, pg. 38, that "Anticipating this result, it follows that even if $u(x)$ vanishes at either or both of the values $x=a$ and $x=b$, both $t^2(a)/u(a)$ and $t^2(b)/u(b)$ still ...
0
votes
0answers
25 views

Continuous embedding of weighted Lebesgue space

Let $w$ belong to the class of Muckenhoupt weight $A_p$ for some $1<p<\infty$ and define the weighted Lebsgue space $$ L^p(\Omega,w):=\left\{u:\Omega\to\mathbb{R} \text{ measurable }: ||u||=\...
2
votes
0answers
12 views

Does the carré du champ operator associated with the generator of a contractive $C^0$-semigroup on a Hilbert space have the diffusion property?

Let $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on a $\mathbb R$-Hilbert space $H$ with generator $(\mathcal D(A),A)$, $\mathcal A$ be a subspace of $\mathcal D(A)$ with $fg\in\...
1
vote
0answers
36 views

Topology textbook for Functional Analysis

Can someone please recommend me an introductory topology textbook written with the functional analysis student in mind? So a book that covers the topology prerequisites a functional analysis student ...
1
vote
0answers
15 views

Semi-linear functional

I'm trying to show that a function defined as $f(x) = \sum_{i=1}^na_ix_i$ on a n-dimensional complex space is semi-linear functional. The additive property is straightforward but the conjugate ...
0
votes
0answers
11 views

Proof that $\psi_{\tau}:=\int_{0}^{\tau}{drU(r)\psi}$ is in a neighborhood of $\psi$

Let $U(t)$ be a unitary one parameter group to show that the family $$\psi_{\tau}:=\int_{0}^{\tau}{drU(r)\psi}$$ is in a neighborhood of $\psi$ Frederic Schuller lecture 12 shows that $$ \lim_{\tau\...
5
votes
1answer
1k views

Are continuous functions strongly measurable?

Measure theory is still quite new to me, and I'm a bit confused about the following. Suppose we have a continuous function $f: I \rightarrow X$, where $I \subset \mathbb{R}$ is a closed interval and $...
2
votes
2answers
51 views

On the Schrodinger fundamental solution

Let $e^{it\Delta}$ be the fundamental Schrodinger solution. If $u_0$ is the corresponding initial data to the problem associated to Schrodinger free equation $u_t = i\Delta u$ and $S(\mathbb{R}^N)$ ...
4
votes
1answer
77 views

Special elements in the $C^*$ algebra $A \otimes \mathcal{K}$.

Context: Let $A$ be an ungraded (not necessarily unital) $C^*$ algebra. $\mathcal{K}$ space of compact bounded operators on an infinite separable graded Hilbert space $H=H_0 \oplus H_1$. Consider ...
0
votes
0answers
11 views

n-times commutator of a function with $-\Delta +v$

Let $f$, $v$ be smooth function and let us assume that $v$ is also bounded. I'm needing somehow a formula for the $n$-times commutator of $f$ (interpreted as multiplication operator) with the ...
2
votes
0answers
32 views

Proving one form of Ito Isometry using Functional Analysis

I would like to know whether it is possible to give a proof of (one form of) Ito Isometry using a tool which I like to call "the functional analysis"-way. Let me explain the settings first. What we ...
0
votes
2answers
35 views

If $U\subset W\subset V$ and $W\cap U^\perp = \{0\}$ then $U=W$

Statement : Let $V$ be a Hilbert space. Let $U\subset W\subset V$ be closed subspaces. Suppose that $W\cap U^\perp = \{0\}$. Then $U=W$. I know this is true in the finite dimensional case (see proof ...