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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28 views

Product topology on a product space of normed spaces is normable iff the product is finite [duplicate]

Suppose $(X_{i}, \Vert \cdot \Vert_i)_{i\in I}$ are all normed spaces over the same field $\Phi= \mathbb{R}, \mathbb{C}$ and suppose $X= \prod_{i \in I} X_i$ is the product space. I want to show that ...
2
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1answer
47 views

Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
1
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1answer
22 views

How to define $E(X)$ when X is a random variable from the sample space to an infinite-dimensional topological vector space?

Let $V$ be an infinite-dimensional vector space with a topology. For simplicity, we can assume $V$ is a Banach space. Let $(\Omega, \mathcal{F}, P)$ be a probability space. Let $X:\Omega \to V$ be a ...
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0answers
22 views

Convolution of a distribution with a $C^{N}$ function.

I've been working on the following Problem from Friedlander's introduction to the theory of distributions: Show that if $u$ is a distribution of order $N$ and with compact support on ...
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1answer
20 views

Closed, convex and balanced subset of a vector space

Let $(X,\|\ \|)$ be a vector space over $K$ and let $B\subseteq X$ be closed, convex and balanced. I want to prove the following: If $x_0\in X\setminus B\Rightarrow\exists\;f\in X^*$ s.t. ...
1
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1answer
24 views

Bounded sequence in $W^{1,p}$ converging to a non-differentiable function in $L^p$

Let $U = B(0,1)$ be the unit ball in $\mathbb R^n$, $p>1$ and $\{u_k \}$ a bounded sequence in $W^{1.p}(U)$. The Rellich-Kondrachov compactness theorem tells us that there is a subsequence $\{ ...
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1answer
60 views

Evaluate for $t\in \mathbb{R}$ $\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx$

Evaluate for $t\in \mathbb{R}$ $$\int_{-\infty}^\infty{e^{itx} \over (1+x^2)^2}dx.$$ Here is what I have done: Let $f(z)={e^{itz}\over (1+z^2)^2}$. This has two poles $z=i$ $z=-i$ and an essential ...
3
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1answer
81 views

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions?

Is it true: If all linear subspaces of a Banach space are closed, then the space is of finite dimensions? My attempt to prove this: For contradiction, suppose $X$ is an ...
0
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1answer
48 views

Show that $e^{\varepsilon |x|^{\varepsilon}}$ grows faster than $\sum_{k=0}^{\infty} {|x|^{2k}}/{(k!)^2}$

I am wondering whether we have for $$f(x):=\sum_{k=0}^{\infty} \frac{|x|^{2k}}{(k!)^2} $$ that $$\lim_{x \rightarrow \infty} \frac{e^{\varepsilon |x|^{\varepsilon}}}{f(x)} = \infty$$ for any ...
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0answers
25 views

Proving that these functionals are bounded, and finding their norms.

Proving that these functionals are bounded, and finding their norms. $$a.)f_1(x):c_o \to \mathbb R , f_1(x)=\sum_{n=1}^{\infty}\frac{x_n}{2^{n-1}} \\ b.)f_2(x):l_1 \to \mathbb R , ...
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1answer
27 views

Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$

Let $B(l^2)$ be the $C^*$algebra of bounded linear operators on the sequence space $l^2$ and denote with $B(l^2)\odot B(l^2)$ the tensor product of $B(l^2)$ with itself, considered as a $*$algebra ...
4
votes
1answer
45 views

Proving that a set of functions is a linear subspace of a vector space

I am attempting to solve the following problem: Let $V$ be the vector space of all continuous functions $f : R → R$ with point-wise addition and scalar multiplication defined. (a) Show that $M_1$ = ...
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1answer
37 views

Sequence in hilbert space, mutually orthogonal vectors

Let $y_1,y_2,\cdots$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1,\cdots,y_n\}$. Assume that $||y_{n+1}||\leq ||y-y_{n+1}||$ for all $y\in V_n$ for $n=1,2,3,\cdots$. Show ...
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0answers
23 views

Banach space and invertible linear operator

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
2
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1answer
35 views

If $H$ is a Hilbert space, are we able to identify the derivative ${\rm D}f(x)$ at some $x\in H$ of a differentiable $f\in H'$ with an element of $H$?

I'm confused about some equation I've seen in a book and want to write down some thoughts. I would appreciate, if somebody could tell me whether I'm terribly mistaken or not: Let ...
2
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1answer
13 views

Is $p\in B(\mathbb{C}^4)$ a s.o.t-limit of a sequence $(a_n\otimes b_n)_{n\in\mathbb{N}}\subseteq B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)$?

Let $L(H)$ the bounded linear operators on a hilbert space $H$. I proved that the inclusion $$i:B(\mathbb{C}^2)\otimes B(\mathbb{C}^2)\hookrightarrow B(\mathbb{C}^4)$$ is not surjective: take ...
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0answers
20 views

Can we derive a Taylor formula for real-valued Fréchet differentiable functions on a normed space?

Using the Lagrange form for the remainder, Taylor's theorem can be stated as follows: Let $I\subseteq\mathbb R$ be an interval, $f\in C^{n+1}(I)$ for some $n\in\mathbb N_0$ and $s,t\in I$ ...
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1answer
13 views

positive operator, projection on Hilbert,$Q|T|Q \ge |QTQ|?$

Let $T$ be an operator on a Hilbert space $H$. And $Q$ be a projection. Whether $$Q|T|Q \ge |QTQ|?$$ Obviously, if $T$ is positive, then $Q|T|Q = |QTQ|$. Also, there are some $T$ such that $QTQ=0$ ...
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0answers
14 views

Following integral is in the domai of the operator or not.

Let $\{T(t):t\geq 0\}$ be a $C_0$ semi group on a Banach space $X$ and let $A:D(A)\to X$ be its infinitesimal generator. We know that for $x\in X$, $\int_{0}^{t}T(s)xds\in D(A).$ Can we conclude ...
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0answers
26 views

Need hint to show that operater is compact

Let H be a separable Hilbert space with the basis $\{e_n\}$. If A is an operator defined by $$Ae=\frac{1}{n} e_n$$. Then show that A is compact. I Just need Hint how to solve such type of problem.
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0answers
9 views

Sequence $\{ e^{2 \pi i n x}\}_{n\in\mathbb N}$ and Paley-Wiener space $PW(0,1)$.

Let us consider the Paley-Wiener space: $$PW(0,1):=\{f\in L^2(\mathbb R)\cap C(\mathbb R),\ \operatorname{supp}\hat f\subset (0,1) \}.$$ Let us consider $\{ e^{2 \pi i n x}\}_{n\in\mathbb N}$, for ...
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2answers
17 views

summing inner product of orthonomal basis

I need some help with some very basic linear algebra when doing calculations in inner product space. Here is a line I got lost when reading a book... \begin{align*} ...
0
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0answers
26 views

Proving that the point spectrum of $T$ is not empty

This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. ...
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0answers
21 views

Approximation of a continuous bounded function on $\mathcal{P}_2(\mathbb{R}^d)$ by Lipschitz functions

Hello everyone I am currently struggling with the following problem. Consider a bounded, measurable and continuous function $f: \mathcal{P}_2(\mathbb{R}^d) \rightarrow \mathbb{R}$ where ...
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0answers
31 views

Eigenfunction basis of Laplacian on a manifold

It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists ...
2
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1answer
21 views

Consequence of the isomorphic relationship between the dual coset & subspace annihilator

Let $(X,\|\ \|)$ be a normed vector space over $K$ and $M\subset X$ be a closed subspace. The annihilator of M is defined as $$ M^{\bot}=\{f\in X^*:f(x)=0\;\;\forall x\in M\}\\ \big(\text{where ...
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0answers
33 views

Find the function $f(x)$ by using its fourier coefficient

It is easy to find the fourier coefficient and fourier expansion of $f(x)$ function. But I want solve the inverse problem How to find the function $f(x)$, if I know its fourier coefficient (or ...
0
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0answers
16 views

Fredholm index in Calkin Algebra

Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space, let $\mathcal{B}\left(\mathcal{H}\right)$ be the Banach algebra of bounded linear operators and ...
3
votes
1answer
48 views

Help in finding a function in $L^p$ such that $f*g=||g||_1 f$ with $g\in L^1(\mathbb{R})$ non negative and fixed

I consider a non negative function $g\in L^1(\mathbb{R})$. I want to find a function $f\in L^p(\mathbb{R})$ such that $$ f*g=||g||_1 \:f ,$$ where $*$ is the convolution. I would be very thankful if ...
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vote
0answers
30 views

$(x_n)$, $(y_n) \in l_{\infty}$, $x_n \geq y_n$, $\forall n \in \mathbb{N}$, $\Rightarrow f((x_n)) \geq f((y_n))$.

Let c = $\{$real sequencies convergent$\}$. We define $\overline{f}: c \longrightarrow \mathbb{R}$, by $\overline{f}((x_n)) = \lim x_n$. We have $\overline{f}$ linear and bounded, for Hahn-Banach, ...
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0answers
24 views

Is this sequence of functions equicontinuous?

For each $n \in \mathbb{N}$, define $f_n: \mathbb{R} \rightarrow \mathbb{R}$ by $f_n(x) = \cos(n+x) + \log\big(1 + \frac{1}{\sqrt{n+2}} \sin^2(n^n x) \big)$. Is the sequence $(f_n)$ equicontinuous? I ...
1
vote
1answer
21 views

Compact Operator with Infinite rank Doesn't have a Closed Image

Let $\mathcal{H}$ be a separable infinite-dimensional Hilbert space. Claim: A compact operator $T$ which has infinite-rank has an image that isn't closed. I'm trying to prove this claim but I'm ...
1
vote
1answer
21 views

open\closed and disjoint sets under R2

I am stuck with the following question: Consider the sets in $\mathbb{R}^2$ defined by $A = \{(x,1/x)| x > 0 \}$, $B = \{(x, −1/x)| x < 0\}$. Prove that the sets are closed and disjoint, and ...
2
votes
3answers
66 views

FUNCTIONS : Theoretical doubt on functions 2

In the functional mathematics language , if i represent function by $$f$$ . What is the theoretical difference between$$f$$ and $$f(x)$$ ? Please provide a lucid explanation.Thanks.
0
votes
2answers
48 views

Transport equation with variable coefficients using characteristics

I want to solve the following pde: $$x\partial_xu(x,y,z)+y\partial_y(x,y,z)+\partial_zu(x,y,z)=0,(x,y,z)\in \mathbb R^3$$ $$u(x,y,0)=u_0(x,y),(x,y)\in\mathbb R^2$$ using characteristics. Until now ...
0
votes
1answer
33 views

Counter example to the parallelogram identity

Let $Z$ be the linear space of all sequences of complex numbers $z=(z_1 , z_2, z_3,..)$ such that $$ \sum^\infty _ {j=1} |z_{2j}|< \infty$$ and $$ \sum^\infty _{k=0} |z_{2k+1}|^2 < \infty$$ It ...
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0answers
53 views

Several questions about the proof of this lemma

I am reading this paper. I have Several questions about the proof of the lemma 3.3. How to Prove 1,2,3,4?
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1answer
22 views

Function of bounded variation and integration

Let f belong to $C[a,b]$. Show that there is a function g that is of bounded variation on [a,b] for which $\int_a^bfdg=||f||_{max}$ and TV(f)=1. This problem appears on page 162 of Royden's Real ...
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1answer
17 views

Given a normed space $X$ and $A:X\to\mathbb R$, how can I compute the second Fréchet derivative of $f(t):=A(x_0+th)$ for some $x_0,h\in X$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a Banach space and $A:X\to\mathbb R$ be Fréchet differentiable, i.e. $\exists{\rm D}A:X\to\mathfrak L(X,\mathbb R)$$^1$ with $$\lim_{\left\|h\right\|\to ...
1
vote
0answers
16 views

semifinite von Neumann algebra, spectral projection, trace

Let $\mathcal{M}$ be a semifinite von Neumann algebra and $\tau$ be a semifinite faithful normal trace on it. Let $T,P_1,P_2\in \mathcal{M}$, where $P_1,P_2$ are projections with $P_1\perp P_2$. Then, ...
1
vote
0answers
20 views

About the Mercer's theorem.

Doesn't the the Mercer's theorem say something stronger than just the spectral theory of compact self-adjoint operators on a Hilbert space applied to the reproducing "kernel" function? As in if I ...
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vote
1answer
16 views

Does a polynomial function on spectrum uniquely define polynomial on operator?

Let $X\subset\mathbb C$ be a compact set, let $T$ be a bounded operator with its spectrum contained in $X$, let $P$ be a polynomial. Is it true that whenever $P=0$ on $X$ then $P(T)=0$?
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vote
1answer
19 views

Find the adjoint of $-\frac{1}{2\pi i}\int_C R_z \ dz$

Let $L$ be a self adjoint linear operator (not necessarily bounded) and $C$ apositively oriented simple closed curve in the resolvent set encircling $\sigma_0 \subset \sigma(L)\subset \mathbb{R}$. ...
0
votes
1answer
18 views

$2$-capacity of a set in $\mathbb{R}^n$

Let $B_r \subset \mathbb{R}^n$ denote the ball of radius $r$ centered at the origin. Consider any set $F \subset B_1$. For all sets $\Omega \subset \mathbb{R}^n$ such that $F \subset \Omega$, define ...
2
votes
2answers
93 views

Application of Banach-Steinhaus theorem

Let $(x_n)$ be a sequence in a Banach space $E$ such that $\sum_{j=1}^{\infty} |\varphi (x_j) |<\infty$, $\forall \varphi \in E'.$ Then $\sup \limits_{\|\varphi\| \leq 1} ...
2
votes
2answers
29 views

Spectrum of an unbounded operator

Consider a densely defined unbounded operator $A_0:D(A_0)(\subset{H})\to H$ which has the following properties: 1- Symmetric, $\langle A_0x,y\rangle=\langle x,A_0y \rangle$ 2- Positive, $\langle ...
-1
votes
1answer
19 views

Can the adjoint of unbounded operators bounded?

Can the adjoint of an unbounded operator be bounded? If not, how to show it? Examples are appreciated. For instance, given an unbounded operator $V: \mathcal{K} \otimes \mathcal{H} \to \mathcal{K} ...
1
vote
1answer
15 views

spectral projection

Let $T$ be a self-adjoint operator on a Hilbert space $H$. $P$ is a projection on $H$. Let $E^{|PTP|}(1,\infty)$ be a spectral projection of $|PTP|$. My question is: whether $E^{|PTP|}(1,\infty) \le ...
1
vote
0answers
28 views

Finding $\|f\|$

Given $f: \ell^2 \rightarrow \mathbb R$ defined by $$f(a_1,a_2,...)=\sum_{n=1}^{\infty} \frac{(-1)^n}{3^{n-1}} a_n$$ Is $f$ a bounded linear functional? If yes, then find $\|f\|$. It is definitely ...
1
vote
0answers
53 views

Integrating $\int _a^b |f(x)| dx$

Is there a way to calculate these without having to sketch out the function first? Seems like you just plus everything when you evaluate the limits. It doesn't seem like it is simply $|g(b)|-|g(a)|$ ...