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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Why is convexity of $S$ needed in these questions?? (Bachman &Narici, Q-17,18)

Let $X$ be a Hilbert Space and let $\{S\}$ be a Convex set in $X$. Let $d=\inf_{x \in S}\|x\|$ . Prove that, if $\{x_n\}$ is a sequence of elements in $S$ such that $\lim_n \|x_n\|=d$, then $\{x_n\}$ ...
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15 views

For what does the formula $(\prod_{t=1}^d[\begin{array}{c}-\frac 12&1&-\frac 12\end{array}]_{l_t,i_t})f$ stand for?

Let $f:\mathbb R\to\mathbb R$ and $$a_{l,i}:=f(x_{l,i})-\frac{f(x_{l-1,(i-1)/2}+f(x_{l-1,(i+1)/2})}2$$ for some $x_{l,i}$. I've read, that we can write $a_{l,i}$ in the following "operator form": ...
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1answer
52 views

$S = \left\{ x^* Ax\mid x \in C^n ,\ x^*x = 1 \right\} \implies S\;$ is compact and convex

Let $\,A \in {\mathbb{C}^{n \times n}}\,$ and $\,S = \left\{ {{x^*}Ax \mid x \in \mathbb C^n,\ {x^*}x = 1} \right\}.\,$ Why is $A$ compact and convex?
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33 views

infinite dimensional hilbert space - uniqueness of series expansion

A function $f(x)$ is expanded in a series of orthonormal functions $$ f(x) = \sum_{n=0}^{\infty} a_n \varphi_n(x) $$ Show that the series expansion is unique for a given set of $\varphi_n(x) $. The ...
2
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0answers
36 views

Weak convergent without completeness implies strong convergence

I want to know if the following holds without completeness: In a normed linear space $H$, $x_n$ is weak convergent to $x$, and $\lim_{n\to\infty} \|x_n\| = \|x\|$ then: ...
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10 views

Spectrum of hypercyclic operators

Can we say the boundary spectrum of (non-quasinolpotent) hypercyclic operators is not included in point spectrum?
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1answer
232 views

Question about proof of regularity of PDE solution in Evans

We can use a Galerkin method to show that there is a solution to a PDE. So suppose $w_j$ is the basis functions. I am interested in regularity of solutions. In the book by Evans, he differentiates a ...
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1answer
24 views

Let $(X,\mathcal{F},\mu)$ be a measure space and let $g\in L^1((X,\mathcal{F},\mu))$.

Let $\phi:[0,1]\to\mathbb{R}$ defined by $$\displaystyle \phi(t)=\int_X \frac{t^3g}{1+t^2g^2}\ \mathsf d\mu$$ Show that $\operatorname{Im}(\phi)\subset\mathbb{R}$ and that $\phi$ is continuous. ...
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1answer
58 views

Part (d) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (d) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
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22 views

What does a functional integral evaluation look like?

I've read the Wikipedia page on functional integration, but it really isn't very easy to understand. There don't seem to be any online videos on the subject either. In addition, when I search online, ...
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30 views

Sufficient condition for a infinite countable or non-countable intersection of open sets is equal to an open set.

Let $(X,\tau)$ a no discrete topological space. If necessary for an affirmative answer consider a metric space $(X, d )$ or a Banach space $(X, \|\,\cdot\, \|)$. In these cases, the topology $\tau $ ...
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2answers
88 views

Surjection of norms

Let $V$ be an infinite dimensional $\mathbb{C}$ (or $\mathbb{R}$) vector space. Suppose there exists two norms on $V$ such that \begin{equation*} \| \cdot\|_1 \leq \| \cdot \|_2. \end{equation*} Is ...
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2answers
33 views

The norm of the extension of an operator

If $T_0$ is the identity operator from $E$ to $E$, where $E$ is a subspace of $F$ and both of them are Banach spaces (maybe not needed). If the bounded linear operator $T$ is an extension of $T_0$ ...
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2answers
1k views

Weakly closed implies sequentially closed

Another problem involving the weak topology: Let $X$ be a normed space and $A \subset X$ weakly closed. Then $A$ is sequentially closed, that is: If $(x_n) \subset A$ and $x_n \xrightarrow{w}x$, then ...
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0answers
45 views

If equality of dual space of a Banach spaces implys the equality of pre-duals?

Assume $ X_1$ and $X_2$ are two Banach Spaces such that $X_1\subset X_2$, i.e., the element belongs to $X_1$ belongs to $X_2$. No assumption on norms. Then I would expect that the dual space of them ...
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1answer
31 views

Definition of space $L^2(\mu)$ where $\mu$ is a Borel probability measure on $\mathbb R$.

Let $\mu$ be a Borel probability measure on $\mathbb R$ with compact support. Consider the space $L^2(\mu)$. It is the first time that I meet this space (usually I have $L^2(\mathbb R)$). Is it still ...
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1answer
52 views

How to prove a set is norm-closed?

I have to prove that the given space is 'norm-closed convex.' I proved the 'convex' part. But I don't know how to prove a set is 'norm-closed' I think I have to do the followings. Let X be a ...
6
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2answers
93 views

What, and how can, topological invariants can be computed from a space's algebra of functions?

The Gelfrand duality says that the category of locally compact Hausdorff spaces (with proper continuous functions) is equivalent to the category of commutative $C^*$ algebras (with proper ...
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1answer
68 views

Prove the Lipschitz constant must be less than 1.

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 ...
3
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1answer
126 views

Proof of the Inverse Function Theorem using the Contraction Mapping Principle.

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 ...
48
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2k views

Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
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21 views

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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37 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 ...
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55 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
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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1answer
25 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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1answer
23 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 ...
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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 ...
2
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1answer
54 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
33 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 ...
2
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1answer
84 views

Hahn-Banach theorem exercise

Let $X$ be a Banach space (over $\mathbb{R}$) and $u,v\in X$ such that $\|u\|=\|v\|=1$ and $\|2u+v\|=\|u-2v\|=3$. Show that there is $f\in X'$ of unit norm such that $f(u)=f(v)=1$. My idea is ...
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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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1answer
26 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 ...
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32 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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1answer
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 $\mathit{A}$,$\mathit{B}$,$\mathit{C}$ ...
3
votes
1answer
70 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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30 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 ...
1
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1answer
27 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 ...
3
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0answers
46 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
19 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 ...
2
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4answers
49 views

Why in the defn of bounded linear functional does the bound depend on $x$?

If $T : X \to Y$ is a linear functional between normed spaces, we say $T$ is bounded if $\exists M > 0$ such that $||T(x)||_{Y} \leq M ||x||_{X}$ for all $x \in X$. Usually, when we say bounded, ...
3
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1answer
64 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

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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1answer
126 views

Is it true that $\int_\Omega u(t,x)\;dx= \left(\int_\Omega u(s,x)\;dx\right)\bigg|_{s=t}$?

Let $u \in L^2(0,T;L^2(\Omega))$. Define $$f(t) = \int_\Omega u(t,x)\;dx$$ and $$g(t) = \left(\int_\Omega u(s,x)\;dx\right)\bigg|_{s=t}$$ for a.e. $t$. Is it true that $f$ and $g$ are the same ...
2
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0answers
20 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 ...
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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2answers
23 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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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 ...
0
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1answer
101 views

finite number of atoms

The space $L^1(\Omega)$ is never reflexive except in the trivial case where $\Omega$ consists of a finite number of atoms—and then $L^1(\Omega)$ is finite-dimensional. this question is in functional ...