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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5 votes
0 answers
72 views

Existence of a postive measurable set such that $T^{-k}(E)\cap E=\emptyset$ for a particular $k\ge 1.$

Let $(X,\mathcal B,\mu)$ be a probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left(\{x\in X: T^n(x)=x\}\right)=0$ for every $n\ge 1.$ Let $A\in \mathcal B$ ...
0 votes
0 answers
23 views

Some questions about integration and operator theory.

As we all know there are multiple integral operators which all basically do the same thing in various contexts. I am talking about operators like the Lebegues integral, Riemann integral and more ...
1 vote
1 answer
71 views

Where is the fallacy?

I'm struggling with an argument that leads to a contradiction, and yet I cannot pinpoint where the fallacy is. Help will be appreciated. Here is the argument: Let $V$ be a normed space and $W \subset \...
3 votes
2 answers
238 views

Exercise with locally convex topological vector spaces

I was trying to solve this exercise from Royden: Let X be a locally convex topological vector space, let $Y \subset X$ be a closed subspace and $x_0 \in X-Y$.Prove that there exists a continuous ...
1 vote
1 answer
20 views

Ask a question on the norm of linear transformation

The below is from a book that I am reading: If $X$ and $Y$ are normed linear spaces we showed in Lemma 4.14 that $B(X, Y)$ is a vector space. We now show that $B(X, Y)$ is also a normed space. While ...
0 votes
0 answers
19 views

Mountain pass theorem when the space is not simply connected

On wikipedia the mountain pass theorem is stated as follows: The assumptions of the theorem are: $I$ is a functional from a Hilbert space $H$ to the reals, $I \in C^1(H, \mathbb{R})$ and $I^{\prime}$...
0 votes
0 answers
16 views

Range Closed and Nonempty Intersection of approximate point spectrum and residual spectrum

I am trying to disprove the statement in $\textrm{(a)(ii)}$. so I need to find a bounded linear operator $A$ with $\operatorname{Im}A$ closed and $\sigma_{ap}(A)\cap\sigma_r(A)\ne\varnothing.$ My ...
0 votes
1 answer
16 views

continuouty of operators

I was given a task to understand, wheter operators $A$ and $B$ are compact, $$\displaystyle A:\ell_2 \rightarrow L_1(\mathbb{R}), (Ax)(t) = \sum\limits_{k=1}^{+\infty}\frac{x(k)}{\cosh^2(kt)},$$ $$B:...
10 votes
1 answer
1k views

Example of an operator with purely residual spectrum

Do you know an example of a linear bounded operator acting on a Banach (or even Hilbert) space whose residual spectrum is non-empty but the point and continuous spectrum are empty?
3 votes
1 answer
105 views
+100

Analysis of an expression involving a function on $\mathbb R^n$. Related to limits, supremums and translations.

Let $n \in \mathbb N, \, 0 < \lambda < n$ and $1 \leqslant p < \infty$, consider the usual Lebesgue measure on $\mathbb R^n$ and define the function $f \colon \mathbb R^n \to \mathbb R$ by $$ ...
0 votes
2 answers
71 views

Example of an integrable function $f$ such that $|\int(1+x^2)f(x)\,dx|<|\int f(x)\,dx|$

For this to be satisfied we need $f\in L^1$, $\int f(x)dx\neq0$, $\int x^2f(x)dx$ to have a different sign to $\int f(x)dx$ and to have smaller absolute value.
0 votes
0 answers
27 views

Bergh-Löfström Theorem 3.2.2

In the following book:https://www.math.chalmers.se/~bergh/Interpolation.pdf, in theorem 3.2.2 is stated that for every $0<\theta<1$ and $1\leq q\leq \infty$ there exist a positive constant non ...
0 votes
0 answers
10 views

Add another Linear Function outside a Closed Subspace, Preserve continuity and linearity?

In an infinite-dimensional Lebesgue space $\mathsf{V}=L^{p}(\mathsf{\Omega },\mathcal{F},\mathrm{P})$, $p\geq1$, on some probability space $(\mathsf{\Omega },\mathcal{F},\mathrm{P})$, there is a ...
3 votes
1 answer
126 views

Collection of all unitary operators in a Hilbert space is a closed set

Let $\mathcal{H}$ be a Hilbert space. $\mathcal{U(H)}$ be the set of all unitary operators on H. Then I want to show that $\mathcal{U(H)}$ is a (norm-) closed subset of the Banach space $\mathcal{L(H)}...
0 votes
1 answer
28 views

Constructing a Continuous Function Below an Increasing Function

Let $f$ be an increasing function defined on $[0,1]$ with $f(0)=0$ and $f(x)>0$ for $x>0$. Does there exists a continuous function $g$ on $[0,1]$ such that $g(x)>0$ on $(0,1]$ and $$g(x)\leq ...
1 vote
1 answer
15 views

Show that $\sup_{x \in A} \vert y(x) \vert = \sup_{x \in A} \vert \text{Re}(y(x)) \vert$ if $A$ is absolutely convex

Let $E$ be a vector space in $\mathbb{C}$. A subset $A \subseteq E$ is absolutely convex if $\lambda x + \mu y \in A \ \forall x, y \in A, \mu, \lambda \in \mathbb{C}$ satisfy $\vert \lambda \vert + \...
0 votes
0 answers
20 views

Weak formulation of the heat equation

Goodmorning We have the heat equation: $$ \begin{cases} \partial_t u - \partial_{xx} u = f(t,x) & \text{in } J \times G \newline u = 0 & \text{on } J \times \partial G \newline u(0,...
6 votes
2 answers
649 views

convergent subsequence in $S$ versus in $\overline{S}$

I am reading equivalent definition of a precompact set $S\subset H$, where $H$ is a Banach space. We have $S\subset H$ is precompact iff every sequence in $S$ has a convergent subsequence. But shouldn'...
4 votes
1 answer
1k views

Adjoint of sum of two operators

Let $A$ be self-adjoint and $B$ symmetric (which means densely defined for me as well) with $A$-bound less than $1$. Does this imply that $(A+iB)^*=A-iB$ ?
2 votes
0 answers
47 views

Prove that this is finite set

Let $(M,g)$ be a Riemannian manifold. Let $\pi_1(M)$ be the fundamental group of $M$. Let $c\in \pi_1(M)$, define $L([c])= inf \lbrace L(\gamma) , \gamma \in [c]\rbrace $. prove that the inf is ...
1 vote
0 answers
20 views

Exercise 4.6 - Topics in Banach space theory (Albiac, Kalton)

I'm trying to solve Exercise 4.6 from Albiac and Kalton's book "Topics in Banach space theory". The exercise is as follows: (a) If $K$ is extremally disconnected, show that for every ...
2 votes
0 answers
32 views

Bounded linear functional with specific values at linearly independent vectors

Let $(E,\|\cdot\|)$ be a normed vector space. Let $\operatorname{dim}(E) = n < \infty$. Given $\{x_1, \dots, x_n\}$ linearly independent vectors and $a_1, \dots, a_n$ scalars we can always ...
0 votes
0 answers
32 views

Existence and continuity of inverse operator

Let $E$ and $F$ be normed spaces. Let $T:E \to F$ is a linear operator and suppose that exists $c>0$ such that $$\|T(x)\| \geq c \|x\|, \quad \forall x \in E.$$ Then it's easy to see that $\...
0 votes
0 answers
33 views

Spectrum of the Laplacian in a Rigged Hilbert Space Defined by Bilateral Laplace Transform

I am currently exploring the spectral properties of the Laplacian $\Delta$ within a rigged Hilbert space $\Phi\subset H\subset \Phi^*$, where $\Phi$ is a subset of $H$, and the function space includes ...
6 votes
1 answer
2k views

Approximate point spectrum of a normal operator

Let $H$ be a Hilbert space and $T:H \to H$ a linear, continuous and normal operator. Then for every $\lambda \in \sigma(T)$ there exists a sequence $(x_n)_{n \in \mathbb N}$ with $\Vert x_n \Vert = 1$ ...
1 vote
0 answers
31 views

Multiplication operator is a closed

Let $\phi$ be a holomorphic function on the unit disk $\mathbb{D}$. We define the multiplication operator $M_{\phi}$ on the following domain $D = \{f \in H^{2}(\mathbb{D}): \phi f \in H^{2}(\mathbb{D})...
1 vote
1 answer
28 views

Upper bound of operator norm of Hadamard (Schur) product

As in Extension of the Schur product theorem to operators, we can define two compact self-adjoint integral operators $A$ and $B$ on $L^2(\Omega)$, by $$ A \phi(x)=\int_{\Omega} a(x, y) \phi(y) d y ; \...
0 votes
2 answers
35 views

Application of Stone's theorem to regular representation

Consider the left regular representation $$\lambda: \mathbb{R}\to U(L^2(\mathbb{R})), \quad (\lambda_x f)(y) = f(y-x), \quad x,y \in \mathbb{R}.$$ By Stone's theorem, there is a positive, self-adjoint,...
1 vote
0 answers
24 views

Spectrum of operator is closure of continuous spectrum

I would greatly appreciate if anybody could look over my proof as I am new to spectral theory. In particular, I am unsure about bounding $A$ and my analysis on the continuous spectrum of $A$. Given $$...
10 votes
1 answer
163 views

Completeness of $X$ with normal distribution $N(\theta,\theta^2)$ with equal mean and standard deviation: Integral of scale of a function

Suppose that $f:\mathbb{R} \to \mathbb{R}$ is a Borel function such that, for every $a > 0$, we always have: $$\begin{equation*} \int_{\mathbb{R}}f(ax)\exp \left(-\frac{1}{2}(x-1)^2\right)dx =0 \...
0 votes
0 answers
27 views

On Extreme points of unit ball in $\mathbb{R}^n$ with respect to the max norm

I was trying to solve the following problem for my non-linear optimization class from the book by Amir Beck. Let $S=\{ x \in \mathbb{R}^n \colon \|x\|_{\infty}\leq 1\}$. Show that $$ext(S)=\{x \in \...
1 vote
2 answers
41 views

Computing the spectrum of the operator $A(f)(t)=tf(t), A:L^1([0,1])\to L^1([0,1])$

Let $A$ be the bounded linear operator from the Banach space $L^1([0,1])$ to $L^1([0,1])$ defined as $A(f)(t) = tf(t)$. I've come to learn that then $A$'s spectrum should be $[0,1]$, but I'm a bit ...
0 votes
0 answers
46 views

Richard Feynman's Definition of Half Derivative [duplicate]

I read from Leonard Mlodinow's Feynman's Rainbow that Feynman has defined the concept of a half-derivative, an operation $g$ on a function $f$ such that $$g(g(f))=f'.$$ I wonder if it is possible to ...
0 votes
1 answer
27 views

Application of Banach-Steinhaus (Uniform Boundness Principle)

Let $E, F$ Banach spaces and $W \subset \mathcal{L}(E,F)$ . If for all $(x,f) \in E \times F' : \sup_{T \in W} |f(Tx)| < \infty$, then $\sup_{T \in W} \lVert T \rVert < \infty $ . I am trying ...
0 votes
0 answers
19 views

Ries representation VS nested hilbert spaces [duplicate]

Consider 2 hilbert spaces V ↪H , V continus dens included in H ,V≠H and the duals H* and V* . As far as the proofs I understand why we sey V↪H=H* ↪V*, but I fail to understand why this example does ...
2 votes
0 answers
24 views

If $h_k\to h$ in $L^2(\mathbb{R}^n$ then $(h_k\ast \varphi_{\varepsilon_k})\to h$ in$L^2(\mathbb{R}^n)$

Let $(h_k)_{k\in \mathbb{R}^n}\subset L^2(\mathbb{R}^n)$ converge to some $h$ in $L^2(\mathbb{R}^n)$ and $(\varphi_{\varepsilon_ k})_{k\in \mathbb{N}}$ be the standard mollification sequence. I want ...
0 votes
2 answers
1k views

Is there any relation between continuity and weak lower semicontinuity?

Let a normed space $S$ be given. Then we say a function $f$ is continuous on $S$ if $s_n\to s$, we have $f(s_n)\to f(s)$, and we say $f$ is weakly $l.s.c.$ if $s_n\to s$ weakly in $S$, then $\...
0 votes
0 answers
46 views

Product basis on $L^2([a,b])$

Suppose we are given an ON basis for $L^2([a,b])$, $\{f_n\}_n$. Then we know that $\{f_nf_m\}_{n,m\in\mathbb{N}}$ is an ON basis for $L^2([a,b]\times[a,b])$ My understanding is that any $g\in L^2([a,...
2 votes
1 answer
32 views

Spectral Permanence Remark in Murphy's C*-algebras

In Murphy's $C^{*}$-algebras book, he states theorem $2.1.11$ which is that if $\mathfrak{B} \subset \mathfrak{A}$ are $C^{*}$-algebras with $\mathfrak{A}$ unital such that $1_{\mathfrak{A}} \in \...
2 votes
2 answers
712 views

Bounded norms on Banach spaces

In Rudin's Functional Analysis text, there's a theorem on page 45: If $\Gamma$ is a collection of continuous linear maps from a F-space space $X$ to a topological vector space $Y$, and if $\{Tx: T \...
0 votes
0 answers
19 views

Why is that $D_n^ku(x',0)=f_k(x')$? (From H\"{o}rmander‘s Book "Linear Partial Differential Operators" (1969) P.55 Theorem 2.5.7)

Here's part of the proof: However, based on my calculations, $D_n^k\hat{u_n}(\xi',0)=\hat{f_k}(\xi')i^k$, rather than $D_n^k\hat{u_n}(\xi',0)=\hat{f_k}(\xi')$. How come $D_n^ku(x',0)=f_k(x')$?
56 votes
2 answers
33k views

When is the image of a linear operator closed?

Let $X$, $Y$ be Banach spaces. Let $T \colon X \to Y$ be a bounded linear operator. Under what circumstances is the image of $T$ closed in $Y$ (except finite-dimensional image). In particular, I ...
0 votes
2 answers
112 views

Show a transformation is Conformal

The question I've been given is as follows A linear map $T:X→Y$ is said to be conformal when it preserves orthogonality; $$∀x,{\tilde x}∈X,〈x,{\tilde x}〉= 0⇐⇒ 〈Tx,T{\tilde x}〉= 0.$$ Show that this is ...
2 votes
2 answers
81 views

There exists a finitely additive measure on $P (\mathbb R)$ such that $\mu([0,1]) = 1$ and it is a translational invariant.

I have the following question: Show that there exists a finitely additive measure $\mu $ on $P (\mathbb R)$ such that $\mu([0,1]) = 1$ and $\mu(A+x) = \mu(A) $ for all $A \subseteq \mathbb R$ and for ...
0 votes
1 answer
33 views

Weak closure of a subset of the unit sphere of $\ell_1$

It is a well-known and standard fact that for every infinite-dimensional Banach space $E$ the weak closure $\overline{S_E}^w$ of the unit sphere $S_E$ of $E$ is equal to the closed unit ball $B_E$ of $...
4 votes
1 answer
961 views

Adjoint Norms in Banach Space

If $T:X\to Y$ is a bounded linear transformation of Banach spaces $X$ and $Y$, then there is an adjoint transformation $Y^*\to X^*$ that satifies $\langle Tx,y^* \rangle =\langle x,T^{*}y^* \rangle$ ...
2 votes
1 answer
64 views

Identity for complex inner product spaces

Suppose $V$ is a complex inner product space. I want to prove that for any $x,y \in V$, we have the following identities: $\langle x, y\rangle = \frac{1}{2\pi} \int_{0}^{2\pi} \|x + e^{it}y\|^2 e^{...
3 votes
1 answer
57 views

Convolution between $f$ and $g$, with $g$ being in the Schwartz class. Does it follow that $f \ast g \in C^\infty$?

Usually, the convolution between two functions $f,g$ defined on $\mathbb R^n$ is given by $$ (f \ast g)(x) = \int_{\mathbb R^n} f(x-y)g(y) \, dy. $$ Right now I am wondering about a specific property ...
0 votes
1 answer
21 views

Orthogonal Projection in an Enlarged Hilbert Space

Let $(H, \langle \cdot, \cdot \rangle_H)$ and $(U,\langle \cdot, \cdot \rangle_U )$ be Hilbert Spaces such that $H$ embeds into $U$. Let $M$ be a closed subspace of $U$, and define $\mathcal{P}$ to be ...
1 vote
0 answers
57 views

Integration by parts with $|\nabla|:=\sqrt{-\Delta}$

What conditions on $f,g$ do I need to justify the integration by parts $$\int f|\nabla|g\,dx=\int(|\nabla|f)g\,dx.$$From $|\nabla|:=\sqrt{-\Delta}$ we have formally that $|\nabla|$ is a self-adjoint ...

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