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).

Filter by
Sorted by
Tagged with
1
vote
0answers
8 views

Notations on Craig-Wayne's 1993 paper on PDE

I'm currently reading Craig-Wayne's Newton's Method and Periodic Solutions of Nonlinear Wave Equations. My background includes undergraduate functional analysis and PDE, and there are some notations ...
0
votes
0answers
7 views

Under what hypotheses is $B$ self-adjoint?

Let $A$ and $B$ be two linear operators such that $B$ is closed and $A\subset B$ with $A$ self-adjoint. It is true that $B$ is also self-adjoint? If it is not, what conditions on $B$ guarantee the ...
0
votes
0answers
14 views

Determine the operator norm of the following function

I have given a function $L: R^2 \to R^2$ , defined as $L(v)= A*v$ where $A$ is a $2\times2$ Matrix \begin{bmatrix} 1 & 0\\ 0 & 2 \end{bmatrix}. Determine the operatornorm and show that a $v$ ...
3
votes
1answer
44 views

why $x_m$ converges weakly to $x_\infty$?

Let $(X,\|.\|)$ be reflexive Banach space and $Y$ be a closed separable subspace of $X$ $\big((Y ,\|.\|)$is clearly a separable reflexive Banach space$\big)$, then the dual space $Y^*$ of $Y$ is ...
0
votes
1answer
29 views

Prove an equivalent norm

Consider $g\in C[0,1]$ a strictly positive function and bounded away from zero, prove $\|f\|_g=(\int_0^1g(x)|f(x)|^2dx)^{\frac{1}{2}}$ is equivalent to $\|f\|=(\int_0^1f^2(x)dx)^{\frac{1}{2}}$. I ...
1
vote
0answers
14 views

Some questions about wave operator.

On sec. 3.4.1 in Schlag & Nakanish: invariant manifolds and dispersive Hamiltionian evolution equations, the authors talked something about wave operators. The wave operators are defined as the ...
2
votes
2answers
15 views

Showing that $G$ is sequentially closed in $X^*$ with the $\sigma(X^*,X)$ topology.

Let be $X$ Banach space. Let $Y$ be a linear subspace of $X^*$ such that $Y$ is dense for $\sigma(X^*,X)$. Let $\sigma(Y)$ be the sigma algebra on $X$ generated by sets of the form $$\{x\in X:(\langle ...
1
vote
0answers
8 views

How it is done the core a first-order differential operator?

Again a question about the core of an operator. I recall the following well known definition. Let $H$ be an Hilbert space and consider $A:D(A)\subset H\longrightarrow H$ a densely-defined, closable ...
1
vote
0answers
8 views

Core of the sum of two densely-defined operators

Let $H$ be an Hilbert space and consider $A:D(A)\subset H\longrightarrow H$ a densely-defined, closable operator. If $\overline{A}$ is self-adjoint, then $A$ is said to be essentially self-adjoint and ...
0
votes
1answer
21 views

Compute the orthogonal complement of a vector space

I have the following question. In the Hilbert space $l^2$, compute the orthogonal complement of the vector space $X=\lbrace x=(x_n)_n\in l^2:x_{2n}=0 \ and \ x_1=x_3\rbrace $.
1
vote
1answer
19 views

Compute the adjoint of an operator in Hilbert space

I have the following question. In the Hilbert space $l^2$, consider the operator $Tx=(\frac{x_n+x_{n+1}}{2})_n$ and $x=(x_n)_n$. Compute the adjoint of operator $T$. I tried to find $T^*$ such that $(...
-2
votes
0answers
23 views

checking out for integrable and differentiability the functional series $f_{n}(x) = \sum_{n=1}^{\infty}{\frac{1}{1+n^2 x^2}}$ in $(\frac{1}{2} ; 1)$

I have functional series $$f_{n}(x) = \sum_{n=1}^{\infty}{\frac{1}{1+n^2 x^2}} ;x \in (\frac{1}{2} ; 1)$$. I have to check it out for integrable and differentiability. I know that $f_{n}(x)$ is ...
1
vote
0answers
17 views

proof that the bilinear form $a(u,v)$ is coercive

I am trying to show that $ a(u,v) = \int_{-1}^{1} \omega u_x v_x dx $ is a coercive and bounded bilinear form to apply the Lax Milgram Theorem. $\omega$ is defined as $\omega(x) := \sqrt{1-x^2}$. I ...
2
votes
2answers
34 views

Lebesgue integral, Is the solution right?

I'am trying to understand Lebesgue integration Compute $\int_{0}^{\pi}$ f(x)dx Where $f(x) = \begin{cases} sin x & \text{ if } x \in \mathbb{I} \\ cosx & \text{ if } x \in \...
0
votes
2answers
20 views

Is the sequence for the harmonic series in the closed unit ball in l^1?

I wish to show that the closed unit ball in $l^1$ is not compact, for which I believe it would be easiest to show that it is not bounded. For this I want to consider the sequence {1, 1/2, 1/3, ... , 1/...
0
votes
0answers
22 views

What is the relation between between the eigenvalues of $M\equiv (I-Q)A$ and the eigenvalues of $MBM$ where $B\equiv QA$?

I have the following spatial operators A, and Q where $$ A \equiv L + C$$ with $L$ being the discrete Laplacian operator and $C$ being the discrete advection operator. The operator $Q$ is given as ...
0
votes
1answer
22 views

Image of a closed unit ball of non-reflexive space

Let $A:c_0\to L_2[0,1]$, $Ax(t)=\displaystyle\sum\limits_{k=1}^\infty\dfrac{x(k)}{ k!+ t}$ -- is linear operator. It is required to prove or disprove the closedness of the set $A(B_1(0))$, where $B_1(...
0
votes
0answers
25 views

Adjoint operator of compact operator is compact

I have the following problem: Given $H$ a Hilbert space, and $T\in\mathcal{L}(H,H)$ a compact operator. Show that $T^*$ is compact, where $T^*$ is the adjoint of $T$. (That is the operator defined by ...
0
votes
1answer
53 views

When $f\left(\|Ax\|\right)\leq \|f(A)x\|$ is true? [closed]

When is the following relation true? $$f\left(\|Ax\|\right)\leq \|f(A)x\|$$ where $A$ is any matrix and $\|\cdot\|$ is the spectral norm. Edit note : The function $𝑓$ is any real concave ...
0
votes
1answer
17 views

Sum of orthogonal projection operators

Let $X$ be an Hilbert space, I am trying to see that if $P$ and $Q$ are orthogonal projection operators then the following are equivalent: $(1) Im Q\subseteq KerP$ $(2)P+Q$ is an orthogonal ...
1
vote
2answers
31 views

Operator norm inequality $\|XY\|\geq\frac{\|X\|}{\|Y^{-1}\|}$

Let $X, Y$ and $Y^{-1}$ be linear operators on a normed space. How to prove the inequality $$\|XY\|\geq\frac{\|X\|}{\|Y^{-1}\|}?$$ I already know that $\|XY\|\leq\|X\|\|Y\|$ but I don't see how I ...
8
votes
2answers
122 views

Complete Metric Space can be a Banach Space?

Let $(S,d)$ be the space of all sequences in $\mathbb{R}$ with the metric $$d(\mathbf{x},\mathbf{y})=\sum_{i=1}^{\infty}\dfrac{1}{2^i}\dfrac{|\xi_i-\eta_i|}{1+|\xi_i-\eta_i|}$$ where $\mathbf{x}=(\...
1
vote
4answers
68 views

Limit $\lim_{\epsilon \rightarrow0} \frac{1}{\sqrt{\epsilon}}\exp\left({-\frac{x^2}{\epsilon}}\right)$ in the sense of distributions [closed]

Compute the following limit: $$\lim_{\epsilon \rightarrow0} \frac{1}{\sqrt{\epsilon}}\exp\left({-\frac{x^2}{\epsilon}}\right)$$ I do not know how the "sense of distributions" is applied in this ...
-1
votes
1answer
40 views

Compactness of the set $S=\{\sin(2^nx)\mid n\in\mathbb{N}\}$.

Let $$S=\{\sin(2^nx)\mid n\in\mathbb{N}\}$$ in the metric space $L^2([-\pi,\pi])$ of Lebesgue square integrable functions. On $[-\pi,\pi]$ with metric defined as $$ d(f,g)=[\int_{-\pi}^{\pi}(f-g)^{2}...
-3
votes
1answer
49 views

Find it adjoint $T^{*}$ [closed]

Show that the operator $$T: \ell_{2} \rightarrow \mathbf{C},Tx := \sum_{n=1}^{n=\infty} \frac{1}{n}x_{n}$$ is bounded.Then find it (Banach) adjoint $T^{*}$.
0
votes
0answers
9 views

Some properties for eigenvectors of the laplacian on Neumann problem

Let $I\subset \mathbb{R}$ and define the eigenvalue problem, \begin{cases} -\Delta w_i = \lambda_i w_i & \text{in } I \\ \nabla w_i = 0 & \text{on } \partial I, \end{cases} The functions $...
1
vote
1answer
22 views

Question on Semigroup theory for evolution equations: Strong continuity of analytic semigroup $e^{-tA}$

I'm studying about Semigroups in Parabolic equations this semester and I'm having a really hard time understanding how these complex line integrals behave from times to times (my complex analysis ...
2
votes
1answer
39 views

weakly convergent sequence converges under a compact operator

Let $H$ be a Hilbert space and $T\in\mathcal{K}(H)$. Show that if $(x_n)_{n\in\mathbb{N}}$ is a sequence in $H$ that converges weakly to $x_0\in H$ then $\lim_{n\to\infty}||Tx_n-Tx_0||=0$. My proof: ...
0
votes
1answer
32 views

Pairing, Hahn-Banach theorem

I'd like to solve the following problem for part (ii). $X^*$ denotes the space of bounded linear functionals on the normed vector space $X$, and $\left<\ ,\right>$ stands for the pairing between ...
0
votes
0answers
25 views

Why do we have the following equality?

Let $X$ be a reflexive space. Let $\phi \in X^{***}$, $\psi\in X^{**}$ and $Q_X:X\to X^{**}$ be defined as $Q_X(x)x^*=x^*(x)$ for $x\in X$ and $x^*\in X^*$. Why do we have $$\psi(\phi\circ Q_X)=(\phi\...
0
votes
1answer
28 views

Infinite dimensional separable Hilbert spaces having an open countable dense subset

While working on infinite dimensional Hilbert spaces, I came up with the following question: under what conditions is the existence of an open dense countable subset assured? If we assume that the ...
1
vote
0answers
39 views

Lebesgue measure when the dimension of Euclidean space approaches infinity

I think it is correct to say that the Lebesgue measure is not defined for infinite-dimensional spaces based on wiki. However, I have related questions: Is the Lebesgue measure well-defined for $\...
0
votes
1answer
26 views

What does the notation $A\in\mathscr{B}(H_1, H_2)$ mean?

I am sorry for the trivial question, but I am a little bit confused about this notation in literature. Let $H_1$ and $H_2$ be two Hilbert spaces. I am interested in understanding what means that an ...
1
vote
1answer
23 views

Which is the spectrum of this operator?

Let $T : \ell_2 \to \ell_2$, $T(x_1,x_2,x_3,...,x_n,...) = (0,\frac{x_1}{1},\frac{x_2}{2},\frac{x_3}{3},..., \frac{x_n}{n},...)$. Which is the spectrum of this operator?
1
vote
0answers
36 views

Showing that operator is compact

Definition: Let $X, Y$ be Banach spaces. Then we call a linear map $T: X \to Y$ a compact operator if $\overline{T(K_1^X(0))}$ is compact in $Y$. $K_1^X(0)$ denotes the ball with radius $1$ and ...
1
vote
0answers
35 views

$n^{1/p}\chi_{[0,1/n]}$ DOES NOT converge weakly to $0$

I'm working on this problem "Let $X=[0,1]$ with the usual measure. For $n=1,2,...$, let $f_n=n^{1/p}\chi_{[0,1/n]}$. Prove that $f_n$ does not converge weakly to $0$ in $L^1([0,1])$." Yes, this ...
0
votes
2answers
56 views

Cardinality of a Hamel basis of $\mathbb{R}^\mathbb{N}$ [closed]

I have the vector space of $\mathbb{R}^{\mathbb{N}}$ with scalars in $\mathbb{R}$. How could I find the cardinal of Hamel's base from $\mathbb{R}^{\mathbb{N}}$?, I can't think of any way to attack ...
0
votes
0answers
26 views

Coset, Hahn-Banach theorem

I would like to solve the following exercise. $X^*$ denotes the space of bounded linear functionals on the normed vector space $X$. Following the hint, we may consider the coset $\{x\}+Y$ of $Y$ ...
0
votes
1answer
25 views

Representation of diagonal operator in $l^2$ as a multiplication operator in $L_2(X,\mu)$ (spectral theorem)

There is a (weaker) version of spectral theorem saying that any self-adjoint operator in Hilbert space is unitarily isomorphic to multiplication operator in $L_2(X,\mu)$, where $(X,\mu)$ is some ...
0
votes
0answers
19 views

Young's inequality to estimate $\int_{\Omega} |u|^{p-2}u v \leq \frac{p-1}{p}\int_{\Omega} |u|^{p} +\frac{1}{p}\int_{\Omega} |v|^{p} $

Let $\Omega \subset \mathbb{R}^{n}$ be a Lebesgue measurable subset and assume $u,v:\Omega \rightarrow \mathbb{R}$ are such that $u,v\in L^{p}(\Omega)$. Let $p>1$. I would like to apply Young's ...
1
vote
0answers
26 views

On a subspace that is isomorphic to a dense subspace

Let $X$ be an infinite dimensional Banach space and let $M$ be a dense subspace of $X$, i.e., $\overline{M}=X$. Let $N$ be another subspace of $X$ such that $N$ is topologically isomorphic to $M$. ...
1
vote
1answer
28 views

Resolvent definition: bounded operator vs. unbounded operator

Maybe my question is obvious in some sense, but I ask here because I didn’t find a “satisfactory” answer on the web. If we have a bounded or unbounded operator, the definition of resolvent changes? ...
2
votes
3answers
80 views

Limit in distributions of $\frac{\sin(tx)}{x}$

How can I find the limit of $\frac{\sin(tx)}{x}$ as $t \to \infty$ in $D'$ ? I understand that i need to see the $\lim_{t \to \infty}{\int_{\infty}^{\infty}{\frac{\sin(tx)\phi(x)}{x}dx}}$ for every ...
0
votes
0answers
44 views

How to construct cyclic vector of diagonal operator $T_\lambda:l_2\to l_2$? [closed]

There is a result saying that diagonal operator $T_\lambda$ is cyclic iff all $\lambda_n$ are distinct. I am looking for an example of cyclic vector of such an operator. Thank you.
0
votes
0answers
19 views

Show that if $A$ is a symmetric operator then its domain is a subset of domain of its adjoint operator

Let $A \in L(D,H)$ and $A$ is symetric operator show that $D \subset D(A^*)$. My attempt: Domain of $A^*$ is defined as follows Let $v\in H$, consider a linear form $\phi_v \in L(D, \mathbb{C}), \ ...
0
votes
0answers
16 views

Biconjugate formula in non-separated locally convex spaces

I have a locally convex space $X$ with topological dual $X^*$ and coupling $\langle x,x^* \rangle:=x^*(x),\ x\in X,\ x^*\in X^*$. For $f:X\to\overline{\mathbb{R}}$ one defines its convex conjugate ...
0
votes
0answers
15 views

Completely continuous operator on discrete time domain [closed]

My question is this, Let us define an operator T on a discrete-time domain. Is it always a completely continuous operator? Thank you in advance.
0
votes
1answer
30 views

Open Mapping Theorem: countexample and example

Consider X the space of the integrative functions at $(1, \infty)$, if $f \in X$, $\|f\|=\int_1^\infty |f(t)|dt$. Let be $T: X \longrightarrow X$ given $T(f)=\frac{f(t)}{t}$. I've already been able ...
3
votes
1answer
48 views

$A\subset B$ with non empty resolvent sets $\implies A=B$?

I read this statement on my notes and I would like that someone help me to understand why it is true. "Let $A$ and $B$ be two linear, closed and densely defined operators such that $A\subset B$. ...
3
votes
1answer
29 views

If $B\subset B(H)$ is a C*-subalgebra and $T\colon B''\to B''$ is linear, bounded and weakly continuous, then $\|T\|=\|T|_{B}\|$

Let $H$ be a Hilbert space and let $B\subset B(H)$ be a C*-subalgebra. Suppose that $T\colon M\to M$ is linear, bounded and operator-weakly continuous, then I want to prove that $\|T\|=\|T|_{B}\|$. ...

1
2 3 4 5
757