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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Closedness of convex sets in a locally convex space

Let $C$ be a convex subset of a locally convex topological vector space. Consider the properties: a) $C$ is closed. b) $C$ is weakly closed. c) $C$ is weakly sequentially closed. d) $C$ is ...
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1answer
12 views

Existence of adjoint operator in Euclidean space

If we define the adjoint operator of linear operator $A:E\to E$, where $E$ is a complex or real Euclidean, $n$- or $\infty$-dimensional, space, as operator $A^\ast:E\to E$ such that $\forall x,y\in ...
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1answer
91 views

Condition on vector-valued function

Does anyone have any ideas on how to show that the following is true: Let $\Omega \subset \mathbb{R}^{n}$ be open and bounded. Consider vector-valued function $$f: \Omega \times \mathbb{R} \times ...
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1answer
39 views

The cone over separable C*-algebra is also separable?

For a C*-algebra $A$, the cone over $A$ is $CA=C_{0}(0,1]\otimes A$ , My question: If $A$ is separable, $CA$ is also separable?
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39 views

Two questions about orthogonal projections on Hilbert space

Let $l_{k}^{2}$ denote the k-dimensional Hilbert space and $\oplus_{1}^{\infty} l_{k}^{2}$ be the infinite direct sum of $l_{k}^{2}$. Let $P_{M}\in ...
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Generalization of the Riesz-Markov theorem

So, my professor mentioned a version of the Riesz-Markov theorem for some kind of general spaces, that yields a maximum and a minimum measure rather than a unique measure (or something along those ...
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1answer
15 views

Is it sufficient to check weak convergence on a (weak* or strongly) dense subset of the dual?

Let X be a Banach space. If $D \subset X^*$ is (weak*ly or strongly?) dense, then does $f(x_n) \to f(x)$ $\forall f \in D$ imply that $x_n \to x$ weakly? My thoughts: If $g_m \to g$ in the dual, then ...
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1answer
20 views

Joint spectrum of $\{a_1,…,a_n\}$

Let $\{a_1,...,a_n\}$ be commuting normal operators on a Hilbert space. Put $A:= C^*(1,a_1,...,a_n)$. By Gelfand theorem ,abelian C*-algebra $A$ is identified with the algebra $C(\Omega)$ of all ...
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1answer
26 views

an inequality on $L_p$ and $l_2$

Let $\{{f_i}\}$ be a countable or finite collection of good functions (e.g. Schwartz functions on $\mathbb{R}$). Let $1<p\le2$. Is it true that ...
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1answer
21 views

adjoint of composition of bounded linear operators

Our lecturer said it shouldn't be any problem to prove this on our own, but I must be missing something obvious! Let $E$, $F$ and $G$ be normed spaces. Let $T \in \mathcal{L}(E,F)$, $S \in ...
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14 views

deduce open mapping theorem from Banach's homomorphism theorem

We have been given this question in our homework, and I really just don't have an idea where to start Use the canonical factorisation to deduce the Open Mapping Theorem from Banach's Homomorphism ...
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28 views

The set of all normal operators on a Hilbert space is not strongly closed

I need an example to show that the set of all normal operators on a Hilbert space is not strongly closed. Also I know that strong operator topology and strong* operator topology coincide in the set of ...
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1answer
29 views

Coercivity definitions

Hi I was given the following definition of coercivity: Let $V$ be a Banach space. The first definition: $A:V \rightarrow V^{*}$ is coercive iff $\exists \zeta: \mathbb{R}^{+} \rightarrow ...
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1answer
43 views

Definition of $C^1, $the vector space of continuously differentiable functions

I asked a question on clarification of the symbol $C^k$. It was confirmed to me that $C^k$ is actually a space of functions. Now my next question in the definition is on $C^0$ and $C^1$. $C^0$ is ...
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2answers
34 views

Notation for the vector space of functions with $k$ continuous derivatives

I saw the following definition given at the mathworld web site: A function with $k$ continuous derivatives is called a $C^k$ function. In order to specify a $C^k$ function on a domain $X$, the ...
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2answers
52 views

Problem 7, Section 2.5 in Kreyszig's Functional Analysis book

Riesz's Lemma, which is 2.5-4 in Erwin Kreyszig's Introductory Functional Analysis With Applications, is as follows: Let $Y$ and $Z$ be subspaces of a normed space $X$ (of any dimension), and ...
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38 views

Characterization of compact subsets in the metric space of all complex-valued sequences

Here's the statement of the Problem 4 after Section 2.5 in Introductory Functional Analysis With Applications by Erwine Kryszeg: Show that for an infinite subset $M$ in the space $s$ to be ...
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1answer
50 views

estimate of infinite norm by $(p,q)$ norms

Let $p$ and $q$ be conjugate exponents, i.e. $\frac{1}{p}+\frac{1}{q}=1$. Prove or disprove: $$ \|f\|_\infty^2\le\|f\|_p\|f'\|_q $$ I think this is true. I tried to prove it using integration by ...
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1answer
52 views

Is this derivative somehow bounded?

I have a function $\phi: \mathbb{R} \rightarrow \mathbb{R}$ that is a test function and $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is defined by $f(x) = \phi(\frac{\|x\|}{n})$. Now if I take any ...
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2answers
58 views

Reals constructed from equivalence classes of Cauchy sequences of rationals.

Is it proper to say (as I keep reading) that the real numbers are "equal" to the equivalence classes of Cauchy sequences in the completion of the rational numbers. Yes, there is a one-to-one ...
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0answers
21 views

About Continuous Functional Calculus

Let $T$ be a compact and normal operator over a Hilbert space and let $\lambda_1,\lambda_2,...$ be the eigenvalues of $T$ with $\lambda_0=0.$ How can I show that for each $u \in ...
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70 views

Solving differential equation by weak formulation and minimizing a functional

I want to give a weak formulation of the boundary value problem \begin{align*} -(c(x)(u'(x)-1))' & = 0 \textrm{ on } \Omega = (-1,1) \\ u(-1) = u(1) & = 0 \end{align*} where $c(x)$ is ...
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1answer
40 views

Approximate $C^{\infty}$ functions by test functions in the Sobolev space norm

I am looking for a way to approximate a function $f \in \mathbb{C}^{\infty} \cap H^m(\mathbb{R}^n)$ by test functions such that I approximate $f$ and all of $f's$ $m-$ derivatives in the canonical ...
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1answer
45 views

Corollary of Banach fixed-point theorem

Let $(X, \left\lVert\cdot\right\rVert)$ be a Banach space. Let $A:X\to X$ be a linear map and $\nu\in \mathbb{N}$ such that $A^k:X\to X$ is a contraction for every $k>\nu$. Is it true that for ...
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2answers
43 views

$f_n \rightarrow 0$ in $L^1$ $\implies \sqrt{f_n} \rightarrow 0$ also?

Let $(X,\Sigma,\mu)$ be a finite measure space, and let $\{f_n : n \in \mathbb{N} \}$ be a sequence of non-negative measurable functions converging in the $L^1$ sense to the zero function. Show that ...
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1answer
71 views

Reiterate Volterra integral operator is a contraction

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 472 here) the statement that Volterra operator $A:L_2[a,b]\to L_2[a,b]$ defined ...
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1answer
27 views

$\ker (I-A)=\{0\}\Rightarrow\text{im }(I-A)=H$ for $A:H\to H$ compact

Let $T$ be the operator defined by $T:=I-A$ where $I:H\to H$ is the identity and $A:H\to H$ is a compact operator defined on Hilbert space $H$. In such a case, if we defined the chain of ...
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1answer
31 views

Riesz Representation Theorem for $l_p$

Let $ 1 \leq p < \infty$, with $q$ the conjugate of $p$, and let $T \in l^{p*}$. Then for some sequence $g \in l^q,$ $T(f)=\sum_{\mathbb{N}} fg$ for all $f \in l^p$. I am trying to prove this ...
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29 views

the spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?
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1answer
30 views

Direct sum of kernel and image of the adjoint operator

Let $H$ be a separable Hilbert space and $T:=I-A$, where $A:H\to H$ is a compact operator. If $T^\ast$ is the adjoint operator of $T$ it can be proved that $\ker T$ and $\text{im } T^\ast:=T^\ast (H)$ ...
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1answer
34 views

Determine whether this map is an isomorphism

Assume all the normed spaces are over $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let $c$ be the space of all convergent sequences equipped with the supremum norm. For $g\in\ell^1$, define the map ...
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1answer
32 views

Convergence in measure of sequence of functions

Hi I don't have a lot of experience in measure theory so that might be basic. If you have a sequence of functions $a_{k}(x): \Omega \rightarrow \mathbb{R}$ such that $$0 \leq\limsup\limits_{k ...
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35 views

A question concerning the Schwartz space

Denote the Schwartz space by $\mathcal S(\mathbb{R})$. I want to show that $\forall n,k \in \mathbb{N} \cup \{0\}$, $\|\cdot\|^{(n,k)} : \mathcal S(\mathbb{R}) \rightarrow [0, \infty)$ defined by ...
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0answers
20 views

Compactness of a sequence

Let $\theta_n(x,t)$ a sequence such that $$\theta_n\rightarrow\theta\;\;\mbox{in}\;\;C((0,T],H^s)\;\;\mbox{where}\;\;s>1.$$ Consider $\phi \in C^{\infty}$ and ...
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1answer
31 views

$\overline{\text{conv}}\{x_n: n\in \mathbb{N}\}$ has empty interior for a w-conv seqeunce $(x_n)_n$.

Given $x_n \to x$ weakly in a Banach space X, $\dim X = \infty$. Show that $\text{int}\left\{ \overline{\text{conv}}\{x_n:n\in\mathbb N \}\right\}$ is empty. Can anyone help me with this problem? ...
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1answer
22 views

Bochner Integral: Approximability

Disclaimer This thread is related to: Bochner Integral: Integrability It is meant to record. See: Answer own Question It is written as jeopardy. Have fun! :) Problem Given a measure space ...
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2answers
50 views

a subspace $X$ is closed iff $X =( X^\perp)^\perp$

let $X$ be a subspace of a Hilbert space $H$. prove: $X$ is closed iff $X =( X^\perp)^\perp$ i do not know how to proceed. any hints would be appreciated. thanks. $X^\perp = ${ $y \in ...
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1answer
20 views

Degenerate Hilbert-Schmidt operators

Let us define a Hilbert Schmidt operator $A:L_2[a,b]\to L_2[a,b]$ by $$A\varphi:=\int_{[a,b]} K(s,t)\varphi(t)d\mu_t$$where $\mu_t$ is the linear Lebesgue measure. A degenerate case is represented by ...
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0answers
25 views

continuous extensions of concave functions

Let $N$ be a lattice. For a ring R we denote $N_R := N \otimes R$. My question is the following: Does a continuous and concave function \begin{eqnarray*} f: N_{\mathbb{Q}} \to \mathbb{R} ...
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0answers
15 views

Discrete time adaption rule

Is it possible to find an update rule for $d(k)$ that satisfy following equation $$\log\frac{d^2(k+1)+1}{d^2(k)+1}=-c\log\left(|f(d(k))|+10\right)$$ where $c>1$ . I appreciate the time you'll take ...
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1answer
33 views

Taylor series for arctan(x)

I am having a serious problem with the following example: $$f(x) = \arctan(x).$$ The task is to calculate the taylor polynom of the third grade. This is easy and I solved it. The second task is to ...
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Show that for every $\epsilon>0$ and every closed ball $K\subset X$ there is an $N$ such that $||T_nx-Tx||<\epsilon$.

Let $T_n\to T$, where $T_n\in B(X,Y)$. Show that for every $\epsilon>0$ and every closed ball $K\subset X$ there is an $N$ such that $||T_nx-Tx||<\epsilon$ for all $n>N$ and all $x\in K$. I ...
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What exactly is a Pettis integral? An example using covariance.

Let $X$ be columns of data with covariance $\Sigma$. It is a fact that $\mathbb{E}(XX^T) = \Sigma$). Is it true that $\Sigma$ is "Pettis integral of $XX^T$? I am trying to understand the nature of ...
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Kinematics(Newtons laws of motion - acceleration) - Practical

Note: initial velocity = 0ms^-1 , (S= displacement) I am trying yo plot a graph of a practical i have earlier done( varying inclination angle and measuring time taken to displace the whole incline), ...
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1answer
34 views

How do find the distribution of a R.V. from this functional/differential equation?

Assume that $X$ is some R.V. with domain $[0,1]$. A function $U:[0,1]\to[0,1]$ is defined as follows: $$U(x)=x\cdot F_X(x)+(1-x)(1-F_X(x)) = 2x\cdot F_X(x)- x - F_X(x) + 1$$ Where $F_X(x)=\int_0^x ...
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1answer
25 views

Hilbert space is sum of closed subspace and its orthogonal

I am trying to solve the following problem: Show that $H = F + F^\perp $ where $H$ is a Hilbert space and $F$ is a closed subspace. Could you give me an idea for how to proceed please? Thanks!
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1answer
126 views

Nonlinear partial differential equations with applications

Has anyone studied the book 'Nonlinear Partial differential equations with applications' by Tomas Roubicek? I am interested in discussing a point of interest in this book. Specifically, on page 52, ...
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0answers
36 views

Computation the different spectrums of an operator

Let $X:=C^0([0,2],\mathbb C)$,$\phi\in X$ and $T\in L(X)$ defined as: $$(Tf)(t):=\phi(t)f(t),t\in [0,2]$$ Compute: $\sigma_p(T),\sigma_c(T),\sigma_r(T),\sigma(T)$ and $\rho(T)$ I am quite new to ...
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1answer
28 views

Measurability of function defined by an integral

Let $A$ be a Hilbert-Schmidt operator defined on $L_2[a,b]$ by $$A(\varphi):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$where $K\in L_2([a,b]^2)$. The fact that $A(\varphi)\in L_2[a,b]$ is showed in the ...
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2answers
25 views

$P_M-P_N$ on which subspace of H is orthogonal projection?

Let M and N are two closed subspaces of Hilbert space H such that $N\subset M$. Also $P_M$ and $P_N$ are orthogonal projections on M and N respectively. It is clear that $P_M-P_N$ is again an ...