Functional analysis is 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, as well as measure, ...

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60 views

Net converging weak-*, implies uniform bound?

Let $E$ be a complex Banach space. A consequence of the uniform boundedness principle is the following. If $(\lambda_n)_{n\geq 1}$, $\lambda$, are elements of $E^*$ such that $$ \lambda_n(x) \...
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51 views

Is there a pseudocontractive mapping that is not strictly pseudocontractive?

Given a Hilbert space $H$, a mapping $T:H\rightarrow H$ is said to be pseudocontractive if $$\|Tx-Ty\|^2\leq \|x-y\|^2+\|(x-Tx)-(y-Ty)\|^2\,\,\, \forall x,y\in H,$$ and it is strictly ...
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23 views

Show that a subset of $(\mathbb R^n,||.||)$ is closed

Let $C$ be a closed subspace of the normed linear space $(\mathbb R^n,\| \cdot \|)$.Let $r(>0)\in \mathbb R$ Define $D:=\{y:\exists x\in C$ such that $\|x-y\|=r\}$. Show that $D$ is closed. My ...
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77 views

Showing this function is continuous $ f:(x,y)\mapsto x^2+y^2$

I have the following function: $$f:\Bbb R^2 \to \Bbb R,\quad f:(x,y)\mapsto x^2+y^2$$ I want to show that this function is continuous by showing that $f^{-1}((a,b))$ is an open set. How do I ...
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1answer
60 views

How to do it by Dominated Conversgence Theorem?

I'm trying to find the limit $$ I = \lim_{n\to\infty} \int_{\mathbb R^d} \frac1{n} |f(x)|^2 x\cdot\nabla\chi (x/n)dx, $$ where $f \in H^1 (\mathbb R^d, \mathbb C)$, $f \in H^2_{loc}(\mathbb R^d, \...
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1answer
53 views

Speed as a function

we were studing the rate of the function $\frac{f{x_1}-f{x_2}}{x_1-x_2}$ if it is positive so the fonction is growing if it is negative so the function is ascending . in this moment our teacher ...
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1answer
40 views

When the singular inner part disappear in inner outer factorization?

I saw this remark in Hoffman's book - "Banach space of analytic function". If $f$ is analytic in a neighborhood of $\bar{\mathbb{D}}$, the closure of $\mathbb{D}$; then in the inner-outer ...
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56 views

Choice of a dense subset of a separable Banach space

I recently came across the following statement, and still can't prove it: Statement: Suppose $X$ is a separable,closed subspace of $L^1(G)$, where $G$ is a locally compact group. Since $X$ is ...
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170 views

Proving $\ell^p$ is complete

Let be $1\leq p\in\mathbb{R}$, denote: $$\ell^p(\mathbb {R})=\left\{(x_n)\subset \mathbb{R}: (x_n) \mbox{ is a sequence with } \displaystyle\sum_{n=1}^{\infty}|x_n|^p<\infty \right\}$$ Prove ...
5
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1answer
122 views

Density of $C^\infty(\mathbb{R}^n)$ in $C^0(\mathbb{R}^n)$

This could be well-known, but I cannot come up with a rigorous proof. I want to prove density of $C^\infty(\mathbb{R}^n)$ in the continuous functions $C^0(\mathbb{R}^n)$ in the following sense: given ...
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0answers
77 views

Sequences and reflexivity

Assume X to be a real reflexive Banach space. Why are sequential topological notions topological notions ? (relatively to the weak topology on X and the weak star topology on X*) For ex : sequentially ...
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38 views

Convergence in the Implicit function theorem?

Possibly a dumb question. In the implicit function theorem we take $F\in C^k(\Lambda\times U, Y)$ with $k\geq 1$, $Y$ is a Banach space, and $\Lambda, U$ are open subsets of Banach spaces $T,X$. If $F(...
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2answers
136 views

Looking for a “job description” for Hölder's inequality

Here's an example of what I mean by "job description" in the post's title: triangle inequality: to be used, whenever the (unsigned) distances between adjacent points in a sequence $x_0, x_1, x_2, \...
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2answers
42 views

A criterion for invertibility of a bounded linear operator.

I'm studying Semigroup Theory and I wasn't able to understand a step in a certain proof. As far as I have been able to understand, the author used the following result: If $A$ is a bounded linear ...
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2answers
157 views

Show that $c_0$ is a Banach space with the norm $\rVert \cdot \lVert_\infty$

Let $ c_0 = \{ x = \{x_n\}_{n \in \mathbb N} \in l^\infty : lim_{n \rightarrow \infty} x_n = 0\}$. Show that $c_0$ is a Banach space with the norm $\rVert \cdot \lVert_\infty$ I am capable of showing ...
2
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1answer
28 views

Equivalence of sequence spaces

Let $m$ be the space of infinite sequences $(x_i), |x_i| \leq 1$ with norm $\sup_{i>0}|x_i|$. Let $\ell$ be the space of infinite sequences $(x_i), \sum_{i> 0}|x_i| \leq 1$ with norm $\sum_{i &...
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23 views

How to analyse the bound of the sum of permutation sequences?

suppose $X=[x_1, x_2, \ldots,x_n]$ ($0<x_1\leq x_2\leq \ldots\leq x_n$), and $$f(X) = \frac{x_1+2x_2+3x_3+\ldots+nx_n}{nx_1+(n-1)x_2+(n-2)x_3+\ldots+x_n}$$ i.e.,$$f(X) = \frac{\sum_{i=1}^{n}{i\...
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1answer
21 views

Open, convex set of TVS

I'm studying LCS using Conway's book. And I had a question about a proof of Proposition 3.2 in chapter 4. The author said, the proof of this proposition is similar to that of proposition 1.14 (If V ...
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4answers
155 views

What's so special about $p=2$ for the $L^p$ spaces?

The Banach space dual of $L^p$ is $L^q$, where $q=\frac{p}{p-1}$, but I don't really understand the motivation behind this. In particular, I find it kind of surprising that the only $L^p$ space whose ...
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4answers
47 views

Monotonicity of $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x >0$.

There is this function I encountered when I was solving a problem and I am trying to study its monotonicity. So the function is $f(x)= \frac{ e^{\alpha x}-1}{\alpha x}$ where $\alpha \geq1 $ and $x &...
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55 views

Is it possible to compare Sobolev space and Polish space?

Is it very easy to say that Sobolev space and Polish space are unrelated? Or we can infer some connection or relation or common structure or generalize one to another? Any comment would be highly ...
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1answer
42 views

A question on Isometry between the orthogonal subspaces of Hilbert spaces

I was reviewing my class-notes on Functional analysis and the professor had mentioned that given a closed proper subspace $U$ of an hilbert space $\mathcal{H}$, $\exists $ a closed subspace $U^{\...
4
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1answer
76 views

Is Riesz measure an extension of product measure?

Suppose $X$ and $Y$ are compact Hausdorff spaces and $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ are finite regular Borel measure spaces. (By regular I mean that every measurable set can be ...
2
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1answer
149 views

spectral theory of Laplacian on $\mathbb R^n$ [duplicate]

Can you describe the spectrum of the Laplacian $ \Delta : H^2(\mathbb R^n) \subset L^2(\mathbb R^n) \rightarrow L^2(\mathbb R^n)$? I am interested for which values $z \in \mathbb C$ the equation $\...
3
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2answers
67 views

What is the range of the operator $T$ I mean I want to determine $R(T)$

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $\ell^\...
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0answers
68 views

Does (L) sets in a dual Banach space X* are weak* precompact? weak* sequentially precompact?

Let $X$ be a Banach space. A subset $B$ of the dual $X$ is said to be $(L)$ set if any weakly null sequence $(x_n)\in X$ converges uniformly to zero on $B$. It is well Known in the theory that ...
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1answer
52 views

Equation in Hilbert space

Solving the following exercise of a list I have: "$H$ is a complex Hilbert space admitting an orthonormal basis $\{e_n\}, n\in \mathbb{N}$ ; $\{\lambda_n\}\subset \mathbb{C}\setminus \{0\}$ is a ...
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1answer
148 views

Is the normalized duality mapping symmetric?

In the area of functional analysis, nonlinear operator theory, the normalized duality mapping on a Banach space $X$ is a set valued map from $J:X\rightarrow 2^{X^*}$ given by \begin{align} J(x)=\{j(...
6
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1answer
153 views

$\sin$ and $\cos$ are the basis of what space?

When learning Fourier expansions, we learn that $\{\sin(mx), \cos(mx)\}_{m \in \Bbb N}$ is an orthonormal basis for our space and thus we can expand functions in it. My question is what space is this ...
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2answers
274 views

Are there any differences between distributions (generalized functions) and probability distributions?

A distribution/generalized function is an element of the dual space of $$S=\{f\in C^{\infty}(\mathbb{R})\colon \|f\|_{\alpha,\beta}<\infty \text{ for all } \alpha ,\beta\}$$ Where $\|f\|_{\alpha,\...
4
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1answer
198 views

Fourier transform not surjective using oppen mapping theorem.

I know that it is possible to prove that the Fourier transform $\displaystyle\mathcal{F}: (L^1(\mathbb R),\|\cdot\|_1) \to (\{f\in C(\mathbb R): \lim_{|x|\to\infty} f(x) = 0\}, \|\cdot\|_\infty)$ is ...
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1answer
30 views

Existence of invariant mean for $\mathbb{R}^n$

I am working on exercise 2.3(b) from "Essential Results of Functional Analysis" by Zimmer and couldn't find a similar question asked here. A mean $m$ on a measure space $(X,\mu)$ is a continuous ...
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1answer
67 views

The dual space of the set of convergent sequences

I'm trying to obtain the dual space of the set of convergent sequences. In proving this, I have to prove some propositions. Let $c$ be the set of convergent complex sequences with $\|\cdot\|_\infty$ ...
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0answers
30 views

The element presentation of convex hull of the union of compact sets

I want to show that the convex hull of the union of compact convex sets $k_{1}$ and $k_{2}$ in a locally convex topology linear space, consist of the points of the form $$ay_1+(1-a)y_2,\ \ \ \ \ y_1\...
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1answer
41 views

What is the difference for the convergence? What is correct?

Suppose we have a Lebesgue integrable function, $f\in L^1(\mathbb{R}^d)$. I would like to approximate it by nice functions for instance $\{f_n\}_{n\geq 1}\subset C_0^\infty(\mathbb{R}^d)$ smooth ...
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1answer
30 views

Two self-adjoint operators with the same eigenvalues and eigenfunctions

How to show two self-adjoint operators (unbounded) on a Hilbert space with the same eigenvalues and eigenfunctions are the same.
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1answer
350 views

Finding operator $A\in L(\ell_1,\ell_2)$ norm

$$X=\ell_1, \ Y=\ell_2, \ A(x_1,x_2,\ldots )=(y_1,y_2,\ldots)$$ Operator is defined as $y_1=x_1,y_n=x_n-x_{n-1}, \ n=2,3,\ldots $ Prove that $A\in L(X,Y)$ and calculate $\|A\|$. First, I check if ...
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1answer
34 views

Banach space operator norm inequality [closed]

$X,Y$ are Banach spaces. $A,B\in L(X,Y)$ and there are $A^{-1},B^{-1}\in L(X,Y)$. Prove that if $$\|B-A\|\leqslant \frac1{2\| A^{-1}\|}$$ then $$\|B^{-1}-A^{-1}\| \leqslant 2\| A^{-1}\|^2 \|B-A\|$$ ...
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1answer
124 views

Identify singularities and classify them. Find the residue of the function at a given point.

Identify singularities of the function $f(z)=\frac{1}{\cos{z^2}}$ and classify them. Find the residue of the function that the point $z_0=\sqrt{\frac{\pi}{2}i}$ . I am hoping to find a clear ...
2
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1answer
46 views

$x\in X$ LCS, $f\in X^\ast$ s.t. $f(x)=1$, $f|_Y=0$

Let $X$ be a locally convex space (topology induced by a family of seminorms $P$ which separates points) and $Y\subset X$ a closed subspace. Assume $x\in X\setminus Y$. Show that there exists a $f\in ...
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2answers
89 views

Why is $(e_n)$ not a basis for $\ell_\infty$?

Let $(e_n)$ (where $ e_n $ has a 1 in the $n$-th place and zeros otherwise) be unit standard vectors of $\ell_\infty$. Why is $(e_n)$ not a basis for $\ell_\infty$? Thanks.
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11 views

Nuclearity of test functions operating on a certain domain

we know of the nuclearity of $D'(R^n)$ and $S'(R^n)$ (distributions/tempered distributions) by using the isomorphy of $D(R^n)$ and the space of rapidly decreasing sequences $s$ (which is nuclear) ...
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1answer
26 views

The functional take its maximal value for $y(t)=-t$

I want to show that the functional $J(y)=\int_0^1 [y'(t) \sin{(\pi y(t))-(t+y(t))^2}]dt$ ,where $y$ is a continuously differentiable function on $[0,1]$, takes its maximal value $\frac{2}{\pi}$ for ...
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0answers
58 views

Cauchy sequence in reproducing kernel Hilbert space

Consider a positive definite kernel $K:\mathbb N\times \mathbb N\rightarrow \mathbb R$. Denote the unique RKHS associated with $K$ by $\mathcal H_K$. The RKHS $\mathcal H_K$ consists of \begin{align} \...
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93 views

$P$ and $Q$ are unitarily equivalent iff dimensions of ranges and kernels are the same

Two projections $P,Q$ are unitarily equivalent if and only if $$dim(randP)=dim(ranQ)$$ $$dim(kerP)=dim(kerQ)$$ How can we show this? One directionn seems easy: If $P$ and $Q$ are unitarily eqv, ...
2
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0answers
62 views

Baire class functions

Lemma: $F \subseteq R$ is both $F_\sigma$ and $G_\delta$ if and only if $1_F$, the indicator function of F, is Baire-class one. Lemma: Let A and B be $F_\sigma$ sets in $\mathbb{R}$. Then there ...
3
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1answer
86 views

Theorem about irreducible representation of $C^*$-algebra

I have been told, that there is a theorem about irreducible representation of $C^*$-algebras, but I have troubles finding it. It is also possible, that this theorem is consequence of some theorem I've ...
2
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0answers
47 views

Non-triviality of Morrey spaces

As is well known, Morrey spaces are widely used to investigate the local behavior of solutions to second order elliptic partial differential equations. Recall that the classical Morrey spaces $\...
2
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0answers
101 views

Do we have a discrete Hilbert space [closed]

Hilbert space and Discrete space are well studied. But many problems appear to be workable in both domain. So do we have a discrete Hilbert space? I mean a usual Hilbert space but from a discrete ...
1
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2answers
36 views

Non-negative functions are a closed subset of $C_b(K)$

Consider the Banach algebra $A=C_{b}(K)$ of all complex-valued bounded continuous functions on a completely regular Hausdorff space $K$ with the supremum norm, and let $C$ be the set $C:=\{g \in A: g(...