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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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
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1answer
47 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
39 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
25 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
36 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
22 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
14 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
15 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
27 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
18 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
28 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
22 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
22 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
34 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 ...
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0answers
17 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
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1answer
49 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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0answers
31 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
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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
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1answer
22 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.
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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
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1answer
34 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 ...
0
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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
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0answers
17 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
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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 ...
1
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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 $P(T)=0$?
1
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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
95 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
32 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 ...
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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
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1answer
16 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 ...
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0answers
30 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 ...
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0answers
54 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 $|g(b)|-|g(a)|$ ...
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2answers
35 views

Using the triangle inequality to prove if $\lim_{n \to \infty} ||x_n -x||=0$ then $\{x_n\}$ is a Cauchy sequence in $X$

Let $X$ be a normed space over the field $\mathbb{K}$. Use the triangle inequality to prove that if a sequence $\{x_n\}$ in $X$ converges in the norm to an element $x \in X$ then $x_n$ is a Cauchy ...
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1answer
21 views

Characterization of support of positive regular Borel measures

Let $\mu$ be a positive Borel measure ona compact Hausdorff topological space. I am trying to prove the following: Show that $x \in support(\mu)$ if and only if $\int_X f d \mu >0$ for every ...
0
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1answer
28 views

Show that $T: l_1 \to l_1$, $T(x) = (a_k\xi_k)_{k \in \mathbb{N}}$ is continuous.

Let $(a_k)_{k \in \mathbb{N}}$ be a sequence such that $$\sum_{k=1}^{\infty}{|a_k\xi_k|} < \infty$$ for all $x=(\xi_k)_{k \in \mathbb{N}} \in l_1$ Show that $T: l_1 \to l_1$, $T(x) = (a_k\xi_k)_{k ...
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0answers
21 views

Dual of sum and intersection spaces of Banach spaces.

If we have complex, compatible Banach spaces $A_0$ and $A_1$, then if $A_0 \cap A_1$ is dense in both $A_0$ and $A_1$ then it should be case that \begin{align*} (A_0 \cap A_1)^* &= A_0^* + A_1^* ...
1
vote
1answer
42 views

The density of $C^1[0,2\pi]$

I am not sure if the inclusion $\{f \in AC[0,2\pi]: f(0)=f(2\pi)=0\}\subseteq \overline{\{f \in C^1[0,2\pi]: f(0)=f(2\pi)=0\}}.$ Here $C^1[0,2\pi]$ is the set of continuously differentiable functions ...
1
vote
1answer
26 views

counter example of sum of closable operators

Let $A$, $B$ and $A+B$ are closable operators. I am not sure the relation $\overline{A+B}\supseteq \overline{A}+\overline{B}$ is true or not(with equality if one operator is bounded. ) And I have a ...
1
vote
1answer
23 views

Neumann Laplacian heat kernel or semigroup representation

I have the equation $$u_t - \Delta u = f\text{ on $\Omega$}$$ $$\partial_\nu u = g\text{ on $\partial\Omega$}$$ $$u(0) = u_0$$ for $f \in L^2(0,T;H^1)$, $g \in L^2(0,T;H^1(\partial\Omega))$ and $u_0 ...
0
votes
0answers
27 views

Approximate SVD

Let $A$ be a complex matrix. A singular value decomposition (SVD) tells us that $A$ can be written as: $$ A = U \Sigma V^*, $$ where $U$ and $V$ are unitary matrices formed by bases of spans of left ...
1
vote
2answers
32 views

positive operator, projection, Hilbert space

Let $T$ be a positive operator on a Hilbert space $H$. Let $P$ be a projection on $H$. Then, it is well-known that $PTP$ is also positive. My question is: whether $T\ge PTP$?
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votes
0answers
21 views

Equivalent conditions for weak $L^p$ spaces for $p\leq 1$

I have difficulty doing the following exercise from Tao's real analysis book: Let $X$ be $\sigma$-finite measure space and $0<p\leq 1$. Then show that the following are equivalent: $f$ is in ...