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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1answer
13 views

Is the set $V=U\cap-U$ balanced?

Let $E$ be a topological vector space and $U$ be an arbitrary neighborhood of $0$. I would like to know if $V=U \cap -U$ is balanced, that is $\lambda V \subset V$ for all $\lambda \in \mathbb{C}$ ...
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4 views

Need help understanding supremum notation in a distance metic on the set of bounded complex sequences.

I'm working on an excersize (from Kreyszig's Functional Analysis book) trying to prove the triangle inequality for a particular distance metric on the set of bounded complex sequences. I think my ...
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29 views

Is H.L. Royden's REAL ANALYSIS, 4th edition, suitable for these two introductory functional analysis courses?

Is the book Real Analysis by H.L. Royden, 4th edition, suitable for two introductory functional analysis courses comprising the following topics? First Course: Metric spaces: A quick review, ...
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2answers
39 views

Let $X= L^2([0,1],\#) $ ( where $\#$ is the measure that counts) $ g\in L^\infty([0,1],\#)$ $A(f)(x)=f(x)g(x)$. Calculate spectrum, eigen values..

Let $X= L^2([0,1],\#) $ ( where $\#$ is the measure that counts) $ g\in L^\infty([0,1],\#)$ and $A \in \mathcal{L} $ as $A(f)(x)=f(x)g(x)$. i)Show $\sigma(A) = \overline{g([0,1])}$ ii) Determine the ...
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2answers
25 views

Zeros of linear combination of basis functions with infinite zeros

Suppose to consider a linear combination $f$ of real functions which are known to have infinitely many zeros on the real line (namely, I am considering the prolate spheroidal wave functions). What can ...
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0answers
25 views

Weak-star topology on probability measures

I'm currently starting to study Robust Statistics (based on Huber's book) and still somewhat struggling with the notion of weak-star topology $\tau_\ast$ on the space of probability measures. Huber ...
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0answers
17 views

Is this the Gateaux differential of $F(u, v)$?

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and suppose $X(\Omega) =X_1(\Omega)\times X_2(\Omega)$ be a Banach space. Moreover let $p, q\geq 1$ be two real numbers. Suppose the functional ...
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0answers
14 views

Powers of (general) closed Operators closed?

I have the following question. Let $X$ be a Banach space (you may specify further properties such as reflexivity or Hilbert space structure if needed) and let $A: \mathcal{D}(A) \to X$ be a closed ...
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1answer
31 views

Compactness without using Heine-Borel in $L^p$ spaces

Consider the set of functions $S=\{\sin(2^nx):n\in\mathbb{N}\}$ in $L^2[-\pi,\pi]$ with the metric $d(f,g)=\left(\int_{\pi}^{\pi}|f(x)-g(x)|^2dx\right)^{\frac1{2}}$. Then is $S$ both closed and ...
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1answer
29 views

A characterization of Hilbert spaces via duality mapping

Let $(X, \|\cdot\|_X)$ is a finite dimensional real normed space and $X^*$ is the dual of $X$. And let $J\colon X\to 2^{X^*}$ be the duality mapping defined by $J(x) = \{f\in X^* : f (x)= \|x\|^2_{X} ...
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37 views

Agmon's inequality on an open subset $\Omega$ of $\mathbb{R}^3$

I'm looking for a reference for what we call the Agmon's inequality on a regular open bounded subset $\Omega$ of $\mathbb{R}^3$ : $$\|u\|_{L^\infty(\Omega)}\leq C \|u\|_{H^1(\Omega)}^{1/2} \|u\|_{H^2(\...
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30 views

If $G$ is a locally compact group equipped with a Haar measure, then $L^{1}(G)$ is a Banach algebra with the convolution as the product operation.

I'm currently working on a research project based on John B. Conway's "A Course in Functional Analysis", specifically Fourier theory on locally compact groups. In his book, Conway claims ...
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1answer
44 views

$A > B$ implies $1 > A^{-1} B$ for operators on Hilbert Spaces

As the title suggests, I have two operators on Hilbert spaces. They are both unbounded but I have bounded the inverse of $A$ (by showing it is strictly positive.) I seek to bound the composition $A^{-...
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1answer
26 views

Orthogonal Complement of quadratic functions in L²([-1,1])

How can we characterize $V^{\perp}$ where $ V= \{v \in L^2([-1,1]): v(x)=ax+bx^2,b\neq 0\}$ ? I've tried looking for $ \{f \in L^2([-1,1]): \int_{[-1,1]}{fv\ d\lambda}=0, \ v(x)=ax+bx^2 \}$ but I ...
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1answer
36 views

Is it possible to work in $L^1$ with a non complete measure?

Basically I think that it is implicit when we traditionally build $L^1(X)$ that the measure is complete. BUT WHAT IF it isn’t complete, is it possible? And what properties do we loss (for sure it won’...
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2answers
60 views

Number of points of discontinuity of $1/\log|x|$

I was solving a few questions from limits continuity and discontinuity when I came across a question asking for the number of points of discontinuity of $f(x)=1/\log|x|$. I could easily observe that ...
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1answer
21 views

If $0 \in \sigma(N)$ and $A = NN^\ast$, then $0 \in \sigma(A)$.

I am going through a longer proof of a theorem which states the following as an intermediate result: Let $N$ be a bounded normal operator on a Hilbert space $H$ with $0 \in \sigma(N)$. Then the self-...
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1answer
19 views

In a metric space, compact implies sequentially compact

I'd like to know if this demonstration is correct. Let $X$ be a metric space and $K \subseteq X$. Show that if $K$ is compact, then $K$ is sequentially compact. $K$ is compact, therefore every open ...
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2answers
26 views

Does $L^1$ imply $L^p$ on finite measure spaces?

If $(\Omega,\mu)$ is a finite measure space, i.e., if $\mu(\Omega)<\infty$, then does $f\in L^1(\Omega)$ imply that $f\in L^p(\Omega)$ for every $p$? This is just a statement that I feel like I've ...
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1answer
57 views

Bounded Linear Operator from $C_0([0,1])$ to $C([0,1])$

Define $$ C_0([0,1]) := \left\{f\in C([0,1]) : \int_0^1f(t)\, \mathrm dt=0\right\}. $$ Show that $T : C_0([0,1]) \to C([0,1])$, given by $$ (Tf)(x) := \int_0^x(t-x) f(t) \, \mathrm dt, $$ defines a ...
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1answer
56 views

A linear map $f: E \to F$ is continuous if and only if $f$ is sequentially continuous.

Let $E$ be an LF-space (see page 126 of $[1]$), $F$ a locally convex space and $f: E \to F$ a linear map. I want to prove that $f$ continuous if and only if it is sequentially continuous. I tried the ...
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1answer
29 views

Compactness of the trace operator in dimension 1

Let $T$ be the operator defined on the Sobolev space $H^1((0,1))$ by $$T:H^1((0,1)) \longrightarrow \mathbb{R} \\ f \mapsto f(0).$$ This operator is clearly a finite rank operator and thus it is ...
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2answers
35 views

Let $X=C[0,1]$ , $A \in \mathcal{L}(X)$, $f \in X$ show that $ \int_{0}^{x} fg$ is compact for any $g \in X$

i)Let $X=C[0,1]$ , $A \in \mathcal{L}(X)$, $f \in X$ show that $ A(f)=\int_{0}^{x} fg$ is compact for any $g \in X$ fixed ii) show that $ 0 \in \sigma(A)$ and see if it belongs to $\sigma_r(A)$ , $\...
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3answers
58 views

Why is $f(t) = e^{ta}$ differentiable in a unital Banach algebra?

Let $A$ be a unital Banach algebra. For $a\in A$, we define $$\exp(a):= \sum_{n=0}^\infty \frac{a^n}{n!}$$ Consider the function $$f: \Bbb{R} \to A: t \mapsto \exp(ta) = \sum_{n=0}^\infty \frac{t^n a^...
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1answer
29 views

Riesz representation theorem vs. natural duality for $L^2$

We know that the spaces $L^p(\Omega)$ and $L^q(\Omega)$ are isometric and isomorphic for $p,q$ conjugate and $p,q \neq 1,\infty$. Call the isomorphism $l\colon L^p(\Omega) \to L^q(\Omega)$. Take $p=q=...
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3answers
43 views

Properties about topological vector spaces

Let $E$ be a topological vector space. First I want to prove that, given a $V \subset E$ balanced and $\lambda>0$ then $$ \lambda V \subset \beta V, \: \forall \;\lambda< \beta. \tag{1}. $$ For ...
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1answer
15 views

Spectrum of an element in the disk algebra.

Consider the disk algebra $A$ of continuous function on the unit disk $D$ that are analytic on the interior of the disk. Is it true that $\sigma_A(f) = f(D)$ for $f \in A$? A simple yes or no suffices....
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0answers
31 views

Can I show this using a contraction semigroup property?

Let $H$ be a (real) Hilbert space, $L$ be an unbounded operator on $H$ with its domain $D(L)$ and $(e^{-tL})_{t\ge 0}$ be a contraction semigroup on $H$. Then, the following holds from a semigroup ...
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1answer
21 views

If $0 \in \sigma_c(A) \bigcup \sigma_r(A) $ then is the linear map $A$ is not open

i) Let $X$ be a Banach space and $A \in \mathcal{L}(X) $ such that $0 \in \sigma(A) $ Show that if $0 \in \sigma_c(A) \bigcup \sigma_r(A) $ then is the linear map $A$ is not open. ii) Examples of ...
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1answer
24 views

extension of integral preserving positive operators on $L^p$ into $L^q$

Let $(\Omega,\mu)$ be a finite measure space. Let $T \colon L^\infty(\Omega) \to L^\infty(\Omega)$ be a weak* continuous contractive positive operator such that $\int_\Omega T(f)=\int_\Omega f$ for ...
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1answer
33 views

continuity of pointwise limit of continuous functions

Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of continuous functions from $[0,1]$ to $\mathbb{R}$, and $\{f_n\}_{n=1}^{\infty}$ converges pointwise to $f$ i.e. $\lim_{n \to \infty} f_n(x) =f(x)$. Define ...
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0answers
62 views

Is the divergence operator surjective?

I'm checking a result that seems to require that the divergence operator $\nabla\cdot$ be surjective from something like the Sobolev space $\textbf{H}^2(\Omega) \cap \textbf{H}_0^1(\Omega)$ onto $H_0^...
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1answer
16 views

Density of certain space in $L^\infty (\mathbb{R},\Sigma, \mu),$ for some finite positive measure $\mu.$

Let $\mu$ be a finite positive measure on $\mathbb{R}.$ Consider the measure space $(\mathbb{R},\Sigma, \mu), $ where $\Sigma$ is the collection of all Borel sets. Q:1) Is it true that the space of ...
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1answer
34 views

Problem in $L^p$ spaces.

$\textbf{Problem:}$ Let $(X,\mathcal{F},\mu)$ a measure space $\sigma$-finite, $p,q \in [1, \infty)$ conjugates and $\vert f \vert < \infty$ $\mu$-a.e. Define for each $n\in \mathbb{N}$ $$ f_n(x) = ...
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0answers
14 views

Approximation by finite subsets and strong resolvent convergence

Let $\mathbb{G}$ be an at most countable set (e.g., $\mathbb{G}=\mathbb{Z}^d$) and $H$ be a self-adjoint operator (not necessarily bounded) on $l^2(\mathbb{G})$. Let $\mathbb{G}_L$ be finite subsets ...
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0answers
29 views

General class of functions satisfying growth condition on a given functional

The question is inspired by Abel Plana summation formula : Is there a general class of functions $f$ that are positive valued on the positive real axis, and which satisfy the following $$\int_0^\...
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0answers
44 views

Proof of Stone's Theorem on unitary groups

I dont understand a particular step in the proof of Stone's Theorem [ B.C. Hall, "Quantum Theory for Mathematicians",p.210-213]. Let me state the Theorem and explain where I got stuck. Stone'...
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1answer
25 views

Closed convex hull of a sequence

I was reading a paper where it was said that for a sequence $(x_n)_n$ of elements in $\ell^1$, the closed, convex hull of the sequence is $$\left\{ \sum_{n \geq 1} \lambda_nx_n : \lambda_n \geq 0 \...
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1answer
26 views

Changing a double integral into a single integral - Volterra-type integral equations

I have a question regarding a calculation that i stumbled upon when proving that a Cauchy problem can be converted in a Volterra-type integral equation. Specifically, this equality: \begin{equation*} \...
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0answers
14 views

The symmetric of a real function with respect to a line

Let consider the function $$f(t)=a t^2+b$$, and the line $D$ defined by $$y=mx+p$$ such that the constants $(a,b,m,p$) are real and $(a\neq 0,b\neq 0,m\neq 0,p\neq 0)$ What is the expression of the ...
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1answer
55 views

if $f$ is $C^1$ do we always have $\|f(y_n)-f(y)\|_{L^2(0,1)} \longrightarrow 0 ?$

Let $f\in C^1(\mathbb{R})$, and $y,y_n\in L^2(0,1)$, assuming that $$\|y_n-y\|_{L^2(0,1)} \longrightarrow 0,$$ can we deduce that $$\|f(y_n)-f(y)\|_{L^2(0,1)} \longrightarrow 0 ?$$ I think it is not ...
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0answers
11 views

How can I prove that $\liminf_{N\rightarrow\infty} d(\eta^N(0),\eta^N(1))\geq 1$?

Let $\eta(s)\in\mathbb{R}^d$ be a smooth function, $s\in[0,1]$, $d\in\mathbb{N}$ is the dimension and $d\geq 3$. Assume $$\int_{0}^{1}|\partial_{ss}\eta^N|ds\leq \frac{C}{N} ~~\text{and}~~ |\partial_{...
3
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2answers
38 views

Restricting a function in the disk algebra

Let $A$ be the disk algebra, i.e. continuous functions on the closed unit disk in $\Bbb{C}$ that are analytic on the interior of the disk. By the maximum-modulus theorem, we have an isometric ...
3
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0answers
60 views

show $\nabla f(\bar x) \geq 0$ and $\nabla f(\bar x)\bar x = 0$.

let $f : \mathbb{R}^n \to \mathbb{R} $ be a convex and differentiable function and $\bar x$ is solution of this problem $$\min f(x) $$ $$s.t \qquad x \geq 0 .$$ Then show $\nabla f(\bar x) \geq 0$ and ...
0
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1answer
46 views

Gaussian with zero mean dense in $L^2$

I have found in this article that linear combinations of Gaussian with fixed variance are dense in $L^2$. Can something similar be true for Gaussian of fixed mean and variable variance? Equivalently, ...
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0answers
45 views

Decomposition of $W^{1,p}(a,b)$ = $W^{1,p}_0(a,b) \oplus E$ where $E$ are affine functions

First of all, I have to prove that for any function $u \in$ $W^{1,p}(a,b)$ there is a unique affine function $ v \in E = \{Ax+B; A,B \in \mathbb{R} \} $ with the same border conditions $ u(a)=v(a) ...
3
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1answer
55 views

How can we characterize weak convergence in $(c, \Vert \, \Vert _{\infty})$?

Let me recall $c = \{ (x_h)_{h \in \mathbb{N}} \subset \mathbb{R} \, | \, \lim_{h \to \infty} x_h = k < \infty \}$ the space of convergent sequences equipped with $\Vert \, \Vert_{\infty}$. What ...
0
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1answer
16 views

Let $X$ be a banach space, and let $U$ be a finite dimensional subspace, then there is a closed subspace $V$ s.t $X=U\bigoplus V$

Let $X$ be a banach space, and let $U$ be a finite dimensional subsapce, then there is a closed subspace $V$ s.t $X=U\bigoplus V$ MY attempt: Let $U=Span\{v_1,...,v_n\}$ and consider the following ...
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2answers
47 views

numerical integration bounded linear operator

$ Let~S,~T_n~:(C[0,1],~||~||_∞)→(R,~|~|)~be~a~linear~operator.$ $S :=\int_{0}^{1}f(x) dx \\$ $T_n := \frac{1}{n}(\frac{1}{2}f(0) + \sum_{k=1}^{n-1} f(\frac{n}{k}) + \frac{1}{2}f(1))$ $Show~that~S,~T_n~...
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2answers
40 views

Finitely spaned subspaces of topological vector spaces [closed]

Im feeling way dumb. Let $F$ be a proper finite dimension subspace of an TVS (with infinite dimension). I have allready studied that $F$ will be closed. And it looks like that "$F$ has empty ...

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