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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Integral of Laplacian eigenfunctions square

The Laplacian densely defined in $L^2(\mathbb{R}^3)$ has eigenfunctions $f_k(x)$ that are defined as generalized functions. I need to define the integral of the square of these eigenfunctions in a ...
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25 views

In every infinite-dimensional TVS, every w-neighborhood of 0 contains an infinite-dimensional subspace (Rudin's FA, p. 66))

In Rudin's Functional Analysis, second edition, p. 66 I bumped into the following proposition: If X is infinte-dimensional [topological vector space with a dual that separates points on X] then every ...
2
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1answer
56 views

Prove there exists a unique local inverse.

I've been set this problem recently and I'm having a lot of trouble with it. Any help would be much appreciated! Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function with continuous derivatives ...
2
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1answer
34 views

No Hilbert space can have countable Hamel basis without using Baire's Category theorem

I have to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...
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2answers
41 views

Given: self-adjoint, monotonic increasing sequence in $L(H)$ such that $\|T_n\|<C$. Why converges $(T_n)$ strongly to a self-adjoint $T\in L(H)$?

Let $H$ be a Hilbert space, $(T_n)\subseteq L(H)$ a sequence such that $T_n^\ast=T_n$ and $T_n\le T_{n+1}$ for all $n\in \mathbb{N}$. There exists a constant $C>0$ such that $\|T_n\|<C$ for all ...
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1answer
25 views

Does $(\ell^{1}(\mathbb Z), \cdot)$ have a bounded approximate identity?

Put $\ell^{1}(\mathbb Z)=\{f:\mathbb Z \to \mathbb C: \|f\|_{\ell^{1}}:=\sum_{n\in \mathbb Z}|f(n)|< \infty \}$ and we note that $\ell^{1}(\mathbb Z)$ is an algebra under pointwise multiplication. ...
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1answer
26 views

Can someone help me to give some hints? Left Hilbert-$C_0(T,K(H))$ module $C_0(T,H)$

I tried to prove example 3.4 from the book Morita Equivalence and Continuous-Trace C$^*$-Algebras by Iain Raeburn and Dana P. Williams, but I get uneasy with notations and ideas. Let me restate my ...
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33 views

Fourier transform of an inverse function.

If for a given function $f(x)$, the Fourier transform is $\hat{f}(p)$; Is there a way to find the Fourier transform of $f(x)^{-1}$ in terms of $\hat{f}(p)$?
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62 views

Assumption in PDE theory

I have an exercise in PDE theory. Let $w \in C^2(U)\cap C(\overline{U})$ where $U$ is open, bounded and connected and $c \in C(\overline{U},\mathbb{R})$ with $c(x) \le 0$ everywhere. Moreover, ...
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32 views

The concept of correlation in functional analysis

I am currently reading a book "measure, integral and probability" by Capinski and Kopp. The correlation between random variables $X$ and $Y$ is defined as the cosine of the angle between $X_c$ and ...
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46 views

Bilinear maps and Bilinear algorithms

How can one intuitively understand the definition of a bilinear map? Is there some way of looking at it geometrically? I found the following definition: Let $\mathit{A}$,$\mathit{B}$,$\mathit{C}$ ...
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1answer
21 views

Is there a text introducing “high order Fréchet derivative” well?

Let $X,Y$ be Banach spaces and $U$ be open in $X$. High-order Fréchet derivatives are defined inductively so that the n-th Fréchet-derivative of a function $F$ is $F^{(n)}:U\rightarrow ...
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47 views

Decomposition of measures acting on sobolev spaces

This is a follow-up question to Decomposition of functionals on sobolev spaces. Let $\Omega \subset \mathbb{R}^n$ be a bounded, open set and $\mu \in H^{-1}(\Omega) = H_0^1(\Omega)^*$. Moreover, let ...
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0answers
20 views

Calculation of a Frechet derivative

Say I have an infinite sequence $X=(x_i)$, $i=1,2,3,\ldots$ such that it's in $\ell^2$ space, i.e. $\sum_{i=1}^\infty|x_i|^2<\infty$. Now, this function that takes this infinite sequence to a real ...
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2answers
38 views

Let $f$ be continuous on $M=A\cup B$, then $f$ is continuous on every $x\in A\cap B$.

Let $M=A\cup B$, a metric space. If $f:M\to N$ is such that $f|A$ and $f|B$ are continuous, then $f$ is continuous in each point $x\in A\cap B$. My approach: If $f:M\to N$, is such that $f|A$ is ...
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1answer
30 views

How to define a “line” and “symmetry w.r.t. a line” in $L_2(\lambda)$ space

For any $x,y\in L_2([0,1],\lambda)$, define the inner product $\langle. , . \rangle$ by \begin{equation} \langle x, y \rangle=\int_{[0,1]} x(t) y(t) \lambda (dt) \end{equation} Is it proper to ...
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1answer
28 views

Notation for subspace of Hölder Space

As mentioned, this is largely a question on notation. I'm reading Fractional Integrals and Derivatives: Theory and Applications by Samko, Kilbas, and Marichev. I'm just starting and I'm curious about ...
3
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38 views

Spectral measures, supports, compact operators

Let $H$ be a Hilbert space, $K:H\rightarrow H$ a compact self-adjoint operator. The spectral measure of $K$ wrt $v\in H$ is uniquely determined by $$\langle K^n v,v \rangle=\int_{\mathbb{R}} x^n ...
3
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1answer
71 views

Existence of a solution to $f(x) = \int_0^1 k(x,y) f(y) dy$

Let $X = (0,1)\times (0,1)$ with the Lebesgue measure, and $k\colon X \to \mathbb{R}$ be a measurable non-negative function such that $$ \int_0^1 k(x,y) dy = 1$$ for every $x \in (0,1)$. My question ...
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21 views

Normal transformation with eigenvalue in real and complex case

It is known that in finite unitary space, due to spectral theorem, for a normal transformation,if the eigenvalues are 1)real 2)positive 3)absolute value 1,then it is 1) self-adjoint 2)positive ...
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56 views

Comparison of Sobolev spaces on an open or closed interval

As noted in my previous question, I am currently working through some books on Sobolev spaces. I am struggling to determine whether, given an interval $I=(0,a)$,the Sobolev spaces $W^{m,p}(I)$ and ...
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1answer
108 views

Sufficient Condition for $f\in L^{1}(\mathbb{R}^{d})$ to belong to $L^{2}(\mathbb{R}^{d})$

Question. Let $\left\{\varphi_{j}\right\}$ be a complete orthonormal system for $L^{2}(\mathbb{R}^{d})$ such that each $\varphi_{j}\in C_{b}(\mathbb{R}^{d})$ (the space of continuous, bounded ...
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2answers
45 views

Examples of algebras that have a bounded approximate identity

We note that $L^{1}(\mathbb R)$ is an algebra with respect to convolution without identity element. However, regarded as a Banach algebra, $(L^{1}(\mathbb R), \ast) $ has a bounded approximate ...
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23 views

Under what hypothesis on the domain is the X-ray transform/John transform operator bounded?

Let $\mathcal{L}$=space of all lines $L$. The X-ray transform is defined here: https://en.wikipedia.org/wiki/X-ray_transform $Xf:\mathcal{L}\to \mathbb{R}$ is defined by: $Xf(L)=\int_{t \in ...
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0answers
9 views

Variational function versus variational solution

I want to minimize the functional $F[f(x)]$ and I'm going to try this in two different ways: First I am going to numerically minimize the functional $F[f(x)]$, leading to the "true solution" $f(x)$. ...
0
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1answer
22 views

Completeness of a system: For all $n$ and within an interval?

Why is the system $\sin((2n-1)x)$ for $n=1,2,\cdots$ complete in $L^2[0,\frac\pi2]$? This means that the Euclidean norm converges for $n=1,2,\cdots$ and for all $x\in[0,\frac\pi2]$ How does one prove ...
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3answers
51 views

Function on $\mathbb Z^2$ whose value equals the average of values at adjacent points $\Rightarrow$ function is constant

This is a reference request. I am not asking for a proof. If I remember correctly, there is a theorem that states that if a bounded [criterion added after editing] function $f:\mathbb Z^2\to\mathbb ...
3
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2answers
29 views

A condition equivalent to equicontinuity

I am doing a problem which is an application of the Arzela-Ascoli theorem, which boils down to proving that a certain condition is equivalent to equicontinuity. Specifically, I am given a sequence ...
3
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1answer
65 views

Multiplication operators on $L^2$

Let $X$ be a $\sigma$-finite measure space, and let $g$ a measurable complex-valued function $X$, which lies in $L^\infty(X)$. I would like to determine sufficient and necessary properties for the ...
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23 views

A is an Hilbert operator, $(A+I)^{-1}$ is continuous with dense domain then A is essentially self-adjoint

We have no information about A, just that is an operator defined in $X\subset H$ where $H$ is a Hilbert space.
4
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1answer
50 views

An mixed weak star convergence problem

Let $\Omega\subset \mathbb R^N$ open bounded. Given a sequence of Radon measure $(\mu_n)$ such that $\mu_n\to \mu$ in weak star sense in $\mathcal M_b(\Omega)$ and $\|\mu_n\|\nearrow \|\mu\|$. Also ...
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0answers
12 views

Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . On the other hand interpolation space which is defined in the wikipedia link: ...
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1answer
171 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
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172 views

Doubts relating to Spaces of type $\mathcal{S}$

I have doubts in the following two questions : 1) What is the Laplace transform of $[x^k\varphi(x)]^{(q)}$, where $\varphi\in \mathcal{S}_\alpha^\beta$ and $-\infty<x<\infty$ , ...
3
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1answer
47 views

Why is the image of a C*-Algebra complete?

I am currently working through the book by Bratteli and Robinson on C* and W* algebras, there is one point at the beginning of chapter 2.3 that is frustrating me. If we take *-morphism to be a ...
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1answer
36 views

Doubts regarding the upper bound for Total Variation

I was studying a chapter on Total Variation & Compactness, where I had gone through the following portion: " We can also relate the total variation with the shifted $L^{1}$-norm. Define: ...
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1answer
124 views

A matrix is positive if and only if it is Hermitian and its eigenvalues are positive [duplicate]

I want to show the equivalence of two definition of positivity. Let $A \in \mathcal{L}(H)$, where $\mathcal{H}$ is the $n-$dimensional Hilbert Space $\mathbb{C}^n$. $A$ is positive if $\langle x,Ax ...
2
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1answer
21 views

If the gradient of $f$ at $x$ has the same direction with $x$ for all $x$, is $f$ radial?

I would like to ask the following question: If $f:% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}\rightarrow% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ is ...
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1answer
50 views

The approximation of supremum of a set.

Given $\Omega \subset \mathbb R^N $ be open and let $g$: $\Omega\to \mathbb R^+$ be a $l.s.c$ function such that $g\geq 1$, but not necessarily bounded above. Also assume that there exists a sequence ...
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3answers
76 views

Is such a multivariate function the product of two univariate functions?

Let $f: \mathbb{N}^2 \rightarrow \mathbb{R}$ be a function of two variables, $f=f(x,y)$. $f$ has the following property: $$ \sum_{y\in A} f(x,y) = 0 $$ where sum on $y$ runs over a fixed ...
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1answer
50 views

$f\mapsto \frac{df}{dx} - \frac{x}{\sqrt{1+x^2}}f $ has closed image and $1$-dimensional cokernel

Let $X$ be the completion of the space of smooth, compactly supported real-valued functions on $\mathbb R$ under the norm $$\|f\|_X^2=\int_{\mathbb R} \left(\frac{df}{dx}\right)^2 + f^2.$$ Let ...
2
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0answers
24 views

Amenability; topology on power sets

I am currently reading the proof of Proposition 2.2. (a direct limit of discrete amenable groups is amenable) in http://people.maths.ox.ac.uk/kar/amenable.pdf . The proof uses the Tychonoff theorem ...
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0answers
64 views

Theorem 6.28 of Rudin's Functional Analysis

I am trying for some time to prove theorem 6.28 of Functional Analysis by Rudin.The theorem says that if Ω is an open subset of $\mathbb{R}^n$ and $Λ\in D^{'}(Ω)$,then there is a family ...
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1answer
40 views

Unconditional basis in $c_0$

We know that in $c_0$ the standard unit vector basis $(e_i)_{i=1}^{\infty}$ is an unconditional basis. For $n\in\mathbb{N}$, let $s_n=\sum\limits_{i=1}^{n}e_i$, my question is that How to prove ...
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4answers
47 views

On characteristic function

Let $X$ be a set and $A,B\subset X$. Can we consider $\mid\chi_A-\chi_B\mid$ as a characteristic function of some subset of $X$? If yes which subset?
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45 views

Subsequences and blocks of Schauder bases

Suppose $X$ is a Banach space and $(e_n)$ and $(f_n)$ are both Schauder bases of $X$. Does there exist a proper closed subspace $Y\subset X$, and appropriate subsequences of $(x_n)$ and $(y_n)$ that ...
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2answers
24 views

Complete eigen-vector basis from non invertible linear application

Consider a non-invertible linear application $O$ acting on a Hilbert space (quantum mechanics). Is there still any chance to find a complete basis of $O$ eigen-vectors or no?
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2answers
48 views

What is meant by “functional analysis is the study of vector spaces endowed with a topology” [closed]

Lecture notes on Functional Analysis by Razvan Gelca open with the definition: Functional analysis is the study of vector spaces endowed with a topology, and of the maps between such spaces. ...
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1answer
37 views

A claim in Krengel's book on Ergodic Theorems.

In Krengel's it's argued that the fact that $\exists 0\ne u \in L_\infty$ orthogonal to $(zI-T)L_1$ , where $z$ is a complex number on the unit circle, $|z|=1$, then $T^* u = zu$. I don't understnad ...
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1answer
19 views

Direct Integral: Measurability

Given a Borel space $\Omega$. Consider plain functions: $$\eta,\vartheta\in\mathcal{F}(\Omega):=\{\eta:\Omega\to\mathbb{C}\}$$ The implication is wrong: ...