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

Extreme points of a subset of dual space of continuous function

$K$ is compact Hausdorff, and $C(K)$ denotes the space of continuous functions on $K$. Let $\mathbb{1}\in C(K)$ denotes the constant function taking value 1, and let $S$ be the subset of $C(K)^*$ ...
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0answers
67 views

An open ball in $C[0, + \infty)$

Consider the space $C[0, +\infty)$ of all continuous, real-valued functions on $[0, + \infty)$ with metric $$ d( \omega_1, \omega_2 ) = \sum_{n=1}^{\infty} \frac{1}{2^n} \max_{t \in [0,n]} ( \min ...
2
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1answer
38 views

positive elements and norm

If $A$ is a abelian $C^∗$-algebra and $a,b$ are elements in $A$ such that $0‎≤‎a‎≤‎1,0‎≤‎b‎≤‎1‎‎$ ‎‎ then $0‎≤‎\|a-b \|≤‎1‎$. My problem is:"Does the same hold if $A$ is not abelian?"
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2answers
41 views

Compactness of a subspace of $L^{2}([0,1])$

Let's consider the following space $K \subset L^{2}[0, 1]$, consisting of fucntions $x(t)$ so that $\sin{t} \leq x(t) \leq t$. How to check, whether this subspace is compact in $L^{2}[0, 1]$ or not. ...
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19 views

Showing the the unit sphere is closed using sequences

Let $(X,\|\cdot\|)$ be a normed space. Prove that every sequence in $S_X=\{x\in X\mid \|x\|=1\}$ converges in $S_X$. My attempt. Let $(x_n)\in S_X$. Then, $\|x_n\|=1$ for all $n$. Now assume that ...
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1answer
66 views

Direct sum of two closed subspaces of Banach space is not closed

I'm looking for an example of two closed subspace of a Banach space (or even a Hilbert space) whose sum is not closed. We have $l^2$ as Banach space and $A$ and $B$ are closed subspaces of $l^2$ : $...
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0answers
13 views

Adjoint of evaluation operator: Inverse Bayesian Analysis

I'm reading "Inverse Problems - A Bayesian Perspective" by Andrew Stuart and I'm stuck with working out an application (an easier form of section 3.2): Consider a random process $u: (0,1) \to \mathbb ...
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0answers
27 views

How to solve the equation $\nabla f=0$ in generalized function by definition

If $f\in\mathscr{D}'(\mathbb{R}^n)$ such that $\forall \phi\in C_0^\infty(\mathbb{R}^n),\forall i,f(\frac{\partial\phi}{\partial x_i})=0$. How can we get the result that $f$ is constant by the ...
2
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0answers
59 views

Norm of a linear functional on $C([0,1])$

I'm trying to calculate norm of the following operator $T:C\left([0,1], \mathbb{K}\right)\rightarrow\mathbb{K}$. (On the domain we have a supremum norm - $\|\cdot\|_{\infty}$.) $T\left(f\right)=\...
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1answer
34 views

Do the holomorphic or meromorphic functions on a domain $D \subseteq \mathbb{C}$ form a Hilbert space $\mathcal{H}$?

In a physics paper I am reading that the meromorphic functions on $\mathbb{C}$ with $f(x) = f(\overline{x})$ form a Hilbert space. $$ \mathcal{H} = \{ f(x) : f(x) = f(\overline{x}) \}$$ Even let $f(...
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1answer
32 views

Unboundedness of multiplication operator

Let $T:\ell^2\rightarrow \mathbb{K}$ be a linear operator given by $T\left(\left\{\lambda_n\right\}_{n=1}^{\infty}\right)=\sum\limits_{n=1}^{\infty}\alpha_n\lambda_n $, where $\{\alpha_n\}_{n=1}^{\...
1
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0answers
18 views

The validity of mollified method to prove the density of $C_0^\infty(\mathbb{R}^n)$

Let $X$ be a function space completed with converge topology (if possible $X$ is normed) such that $C_0^\infty(\mathbb{R}^n)\hookrightarrow X\hookrightarrow L^1(\mathbb{R}^n)$ is continuous dense ...
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1answer
23 views

What is the n-concavification of a Banach space?

I'm reading this paper about polynomials in Banach spaces and the authors use the notion of the n-concavification of a Banach space $X$ It is the first time that I encounter this concept. What is it? ...
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1answer
57 views

Calculating norm of an operator

Let $X=C\left([0,1]\right)$ be equipped with norm $\|f\|:=\sqrt{\int\limits_0^1 |f(t)|^2 dt \ + \sum\limits_{n=1}^\infty 2^{-n} |f(x_n)|^2}$, where $\{x_n\}_{n=1}^\infty \subset[0,1]$, s. th. $x_n\...
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1answer
60 views

Proving that a given finite-dimensional vector space is isometrically isomorphic to $(\mathbb R^n,\|\cdot\|_\infty)$

Let $X$ be an $n$-dimensional vector space with a basis $\{e_1,\dots,e_n\}$. Consider the norm $\|\sum_{i=1}^n \alpha_ie_i\|=\max_{i\leqslant n} |\alpha_i|$ for $x=\sum_{i=1}^n\alpha_ie_i\in X$. We ...
3
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1answer
69 views

Spectrum of right shift operator in weighted $l2$ sequence space

Let $l_2(a)$ be a hilbert space defined with following inner product: $\langle x_n,y_n\rangle = \sum a^k x_k y_k$. (It's a weighted sequence space with the weights $\omega_i = a^i$). It's elements ...
6
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1answer
73 views

How to proof $C_0^\infty(\mathbb{R}^n)$ is dense in $H^s(\mathbb{R}^n)$ by using mollifier

Since the definition of $u\in H^s(\mathbb{R}^n)$ is $\left(1+|\lambda|^2\right)^{s/2}\hat{u}(\lambda)\in L^2(\mathbb{R}^n)$ I find it difficult to give an constructive prove that use mollifier. let $...
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1answer
31 views

Sobolev function and polynomials

Let $\Omega\subset\mathbb{R}^2$ be a bounded open set with Lipschitz boundary. Let $v\in H^{l+1}(\Omega)$. I have to prove that There exists a unique $p\in\mathbb{P}_l$ such that $$ \int_\Omega D^\...
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1answer
24 views

Show spectrum for operator $S$ and $T$ coincide given that there is some $R$ s.t $T=R^{-1} S R$

Im really stuck on the following excersice in functional analysis. Show spectrum for operator $S$ on some Banach space $Y$ and $T$ on $X$ coincide given that there is some inverable $R$ s.t $T=R^{-1}...
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1answer
32 views

Property of conditional expectation operator in $L^1$.

Let $\mathscr{G}\subset\mathscr{F}$ be two $\sigma$-algebras. It's easy to see that the conditional expectation operator $$E[\,\cdot \mid\mathscr{G}]\in \mathscr{L}(L^1(\mathscr{F}))$$ satisfies $\|E[\...
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3answers
208 views

The Schwartz space is dense in $L^p$

Is there any hint to prove that for every $1 \le p < \infty $ the Schwartz space is dense in $L^p$? Thanks so much.
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1answer
32 views

Relation betwen dimension of Hilbert space and cardinality of its dense subset

Suppose $H$ is infinite dimensional Hilbert space. Let $A$ be a dense subset of $H$. How to prove that $\mathrm{dim}_{\mathrm{orth.}}\ H \le \mathrm{card}(A)$ ? When we have equality ? I need only ...
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1answer
33 views

positive elements of $C^*$-algebra

If $A$ is a abelian $C^*$-algebra and $a,b$ are elements in $A$ such that $0‎\leq ‎a‎\leq ‎1,0‎\leq ‎b‎\leq ‎1‎‎$ then $0‎\leq ‎a‎b\leq ‎1‎$. My problem is:" Is it true if $A$ is not abelian?"
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1answer
47 views

“Trivial” extension of functional bounded?

Suppose I am in $\ell^{\infty}$ and define the $\lim$ functional on the subspace of convergent sequences. Is the "trivial" extension which is zero on all non convergent sequences a bounded functional ...
1
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1answer
55 views

States on a $C^*$-algebra

I know that if $A$ is a non-zero and unital $C^*$-algebra then $S(A)$ (the set of states on it) is weak${}^*$ compact. My problem is: Does the same hold if $A$ is not unital?
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1answer
74 views

The complement of a first category set in X is a set of second category.

Let X be a complete metric space. Then the complement of a first category set in X is a set of second category in X. What is explain in my class is "if the complement of a first category set is a set ...
2
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1answer
42 views

The k-th derivative of the resolvent set

I want to prove $$\frac{d^{k}}{dz^{k}}(zI-A)^{-1}=(-1)^{k}k!(zI-A)^{-k-1}$$ I have the resolvent equation $(zI-A)^{-1}-(\lambda I-A)^{1}=(\lambda-z)(zI-A)^{-1}(\lambda I-A)^{-1}$, i.e. $$\begin{...
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1answer
37 views

Why is a von Neumann algebra is closed with respect to weak * topology?

I was trying to prove that the identity map between a von Neumann algebra $(A,\mbox{ultra weak topology})$ with respect to ultra weak topology and the von Neumann algebra $A$ with respect to weak* ...
3
votes
5answers
146 views

Picard's existence theorem, successive approximations and the global solution

Picard's existence theorem states that if $U$ is an open subset of $\mathbb{R}^2$ and $f$ is a continuous function on $U$ that is Lipschitz continuous with respect to the second variable then there is ...
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2answers
117 views

Support of a distribution, what does it mean?

In my course notes the support of a distribution (continous lineair functional) is defined as follows: Definitions First it defines something like open annihilation sets: An open annihilation ...
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2answers
32 views

prove that $F$ is a closed set and $p(F)$ is not closed

Take a projection $\Bbb R\times\Bbb R\to\Bbb R$ which is given by $p(x,y)=x$. So, if you choose the closed set $F=\{(x,y)\mid xy=1\}$ then $p(F)$ is not closed. why $F$ is closed and why $p(F)$ is ...
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0answers
43 views

Interpretation of $L_2 ([-\pi , \pi])$

What is the interpretation of $L_2 ([-\pi , \pi])$ in laymans terms? How is $L_2 ([-\pi , \pi])$ different to $L_1 ([-\pi , \pi])$? Does $L_2 ([-\pi , \pi])$ just mean that the function is twice ...
0
votes
2answers
55 views

Prove that $A$ is closed set

Define $f_n$ as: $$f_n(x)=\begin{cases}1-\frac{1}{n}, & \text{ if } \frac1n\le x \le 1\\0, & \text{ if } x=0\end{cases}$$ and linear on the rest (but so that the function is continuous, ...
2
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0answers
24 views

How does one integrate $e^{iS(q)}$with respect to $q(t)$?

Let $q(t)$ be some function. Consider the integral $$\int e^{iS(q)} Dq(t)$$ For example, suppose $S(q) = q^2$ Or suppose $$S(q) = \int_0^T \frac{1}{2} m\dot{q}^2$$ Or suppose $$S(q) = \int_0^T \...
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1answer
28 views

computing an operator norm exercise

Let $X=\{x=(x_1,x_2,...):x_i\in\mathbb{R}$, $x_i=0$ for almost all $i\in\mathbb{N}\}$ with the norm $\|x\|=\sup_{i\in\mathbb{N}}|x_i|$. Let $S:X\to X$ be a linear operator defined by $Sx=(x_1+x_2+x_3+....
8
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0answers
121 views

Fast convergence in $L^1$ implies convergence almost everywhere

This is a proof-verification request. Claim: Let $(X,\mathscr M,\mu)$ be a measure space. Let $f_n$ ($n\in\mathbb N$) and $f$ be measurable, integrable, real-valued functions such that $(f_n)_{n\...
2
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2answers
30 views

Examples of product of two $L^1_\text{loc}$ functions that is not $L^1_{\text{loc}}$

Let $f\in L^1_\text{loc}$ and $g\in L^1_\text{loc}$, does $fg \in L^1_\text{loc}$? My textbook says it isn't in a general case. However if $g\in \mathcal{E} = \mathscr{C}^\infty$, then $fg\in L^1_\...
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1answer
47 views

Incompleteness of $\ell^1$ with respect to $\sup$ norm

I'm trying to make an example that shows $\ell^1$, that is the space of complex sequences that the sum of the norms of their components is finite, is not complete with respect to $\sup$ norm. And ...
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1answer
46 views

Find on $C[0,1]$ closed and bounded set $A$ that there are no such $f,g \in A$ that imply $\operatorname{diam}(A)=d(f,g)$

Find in $C[0,1]$ closed and bounded set $A$ such that there are no $f,g \in A$ that imply $\operatorname{diam}(A)=d(f,g)$, where $\operatorname{diam}(A) = \sup\{d(f,g)\mid f,g \in A\}$.
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1answer
32 views

How to handle $\lim_{h \rightarrow 0} \frac{1}{h} \int_{0}^h f(s) ds$

In my lecture notes I found the following identity: $$ \lim_{h \rightarrow 0} \frac{1}{h} \int_{0}^h f(s) ds = f(0)$$ without any explanation on what's going on here. I think in the lecture we said ...
0
votes
1answer
25 views

Show that $\overline{[S]}^{\bot}=S^{\bot}$, for any subset of a inner product space.

I have done a solution for this problem: Let $X$ be an inner product space and $S$ a subset of $X$. Show that $\overline{[S]}^{\bot}=S^{\bot}$. But I am note sure that my solution is correct. If ...
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1answer
67 views

Why does the weak topology make the functions in $E'$ continuous?

Let $E$ be a Hausdorff locally convex topological vectorspace. Consider the source $p_f\colon E\to\mathbb{R}$, where $p_f(x)=\lvert f(x)\rvert$ and $f\in E'$, the continuous dual of $E$. The maps $p_f$...
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1answer
34 views

If $Y$ is a proper closed subspace of a Hilbert space $H$, how do you prove the existence of some $f \in H'$ with the properties below?

If $x_0 \in H - Y$ and $\delta = \inf_{y \in Y} \|y - x_0\|$ then there exists $f \in H'$ such that $$\|f\|=1, \quad f(y)=0 \,\,\text{for all} \,\,y\in Y, \quad f(x_0)=\delta .$$ So far all I have is ...
2
votes
1answer
63 views

What is the dual of the disc algebra viewed as a Banach space?

Let $A$ be the disc algebra, i.e., $A=\{f\in C(\bar{U}):f \text{ is holomorphic in }U\}$, where $U$ is the unit disc in the complex plane. The norm considered is the supremum norm. Are there any ...
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1answer
26 views

Help needed with understanding the proof of “If $X'$ is separable, then $X$ is separable.”

I may be forgetting something trivial here, but I don't follow (or understand the reasoning for) how the contradiction to the density of $(f_n)$ in $U'$ is achieved? I would like for someone to ...
1
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0answers
54 views

Dual space of $l^p$ is $l^q$.

I was reading functional analysis from Kreyszig. While proving that dual space of $l^p$ is $l^q$, I came across a doubt. I have attached the screenshot. In this they are applying f to $x_n$ but for ...
1
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1answer
49 views

Operator norm of matrix

Consider $\mathbb{R}^2$ with norm $\Vert (x,y) \Vert=\sqrt{x^2+y^2}.$ I would like to compute the operator norm w.r.t. the above norm of a matrix $$A=\begin{pmatrix} a_{11} & a_{12} \\ a_{21} &...
0
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0answers
43 views

Confusion about Theorem structure of Structure theorem for Gaussian measures

Structure theorem for Gaussian measures is explained in the book by Fritz P. & D. Victoir "Multidimensional Stochastic process as a rough paths" on page 606. I have some confusion as illustrated ...
2
votes
1answer
56 views

Easy examples of the Arzela-Ascoli Theorem

Let $X$ be a compact metric space. $M \subseteq C(X)$ is relatively compact if and only if $M$ (i.e. its elements) is equicontinuous and uniformly bounded. I've been told that this theorem gives me ...
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1answer
24 views

Unbounded operators: product of adjoints strictly extended by the adjoint of product

It is well known that, if $T,S$ and $ST$ are densely defined operators on a Hilbert space $H$, then $T^* S^* \subset (ST)^*$. The proof of this is easy. Moreover, it's readily seen that equality ...