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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bounded linear operators is B(X,Y) is complete if Y is complete

QUESTION#1 is why he required that the space Y is complete not only the range of the operators is complete? QUESTION#2 In the proof of this theorem : i take a cauchy sequence {Tn} from B(X,Y) and ...
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84 views

Question about proof of Browder, Minty Theorem

Could someone please assist with the following question: In the following SET OF NOTES, I am interested to know how the author obtains "By Lemma 1.11, the Galerkin equations (2.5 has a solution ...
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43 views

Orthogonal basis in infinite dimensional spaces

It is a well known fact that a symmetric bilinear form $g$ on a finite-dimensional vector space $V$ over a field $F$ of characteristic $0$ admits a orthogonal basis $\{e_i\}$, i.e. $\{e_i\}$ is a ...
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23 views

Dense domain of an operator

Suppose that $T$ is a (possible unbounded) self-adjoint operator on a Hilbert space $H$, thus the domain $D(T)$ of $T$ is dense in $H$ and the graph of $T$ is closed in $H\times H$. I want to prove ...
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77 views

Greatest open ball of invertible elements in a Banach algebra

Let $a$ be an invertible element of a Banach algebra $A$. Then we know that also each $a+b$ with $b\in A$ and $||b||<||a^{-1}||^{-1}$ is invertible. Now my question is whether ...
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Does a set of 'm' linearly independent continuous functions constitute a Hilbert Space

If I have a Sobolev space $\mathcal{H}^m[a,b]$ of functions $f : [a,b]\rightarrow\mathbb{R}$ where for all $f \in\mathcal{H}^m[a,b]$, $f$ and all derivatives up to order $m-1$ are absolutely ...
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37 views

Show an operator is compact if $\sum \|Te_n\| < \infty$

Let $H$ be a separable Hilbert space, define a bounded linear operator $T:H \rightarrow H$, show it is compact if $\sum \|Te_n\|_H < \infty$. My attempt: We show that $T(B)$ is totally ...
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35 views

Non-separable Hilbert spaces in duals

A topological space $X$ satisfies the countable chain condition if every family of pairwise disjoint open sets in $X$ is countable. I am looking for a reference to the following fact: Suppose that ...
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30 views

Spectrum of integral operator

Given $g\in C^1([0,1]\times[0,1])$, consider the operator $$Tu(x) = \int_0^1 g(x,t) u(t) dt$$ defined on $u\in C([0,1])$. Discuss the spectrum of T. My attempt: First I can show that $T$ is ...
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26 views

How to show an operator is invertible?

How to show an operator is invertible in an abstact setting? I only know that if $\|T\|< 1$ then $I-T$ is invertible. For example: Let $(X_1, \|\cdot\|_1)$ and $(X_2, \|\cdot\|_2)$ be Banach ...
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52 views

Urysohn's lemma with Lipschitz functions

In a complete and separable metric space $(X,\mathrm{d})$ given an open set $U$ and a closed set $K\subset U$. Is it possible to find a Lipschitz function $f$ such that $f|_K=1$ and $f|_{X\setminus ...
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39 views

Weak convergence exercise.

Let $(f_n)$ be a sequence in $L^2(\mathbb R)$ and let $f\in L^2(\mathbb R)$ and $g\in L^1(\mathbb R)$. Suppose that \begin{eqnarray*} f_n\rightharpoonup f \hbox{ weakly in }L^2(\mathbb R)\,, \\ ...
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33 views

Show that if X is a normed linear space, then any finite-dimensional subspace M of X must be closed.

It suffices to show that any proper subspace M of X is closed, since if M is not proper the result is trivial. I am unsure how to approach this proof. Contradiction seems a little messy, as supposing ...
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25 views

Soft Question: What are some elementary motivations of using functional analysis to study probability theory?

Recently i've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure theoretic ...
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19 views

Function smoothing using convolution

I have a function $\hat f$ which is an estimator of an unknown function $f$. The estimator $\hat f$ looks pretty irregular (see the red line). I would like to smooth it with some kernel function ...
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29 views

Approximating by smooth functions with compact support.

Consider a bounded domain $D \subset \mathbb{R}^n$ and the Sobolev space $H^1_{0}(D):=\overline{C_c^{\infty}(D)}^{W^{1, 2}(D)}$. Further, consider a Sobolev function which happens to be smooth: $u\in ...
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33 views

Given $A_n : X\rightarrow Y$ linear and continuous, for each $x\in X$ $A_n(x) \rightharpoonup A(x)$ in $Y$, is $A$ continuous?

Given $A_n : X\rightarrow B$, a linear and continuous operator between two Banach spaces, for each $x\in X$ $A_n(x) \rightharpoonup A(x)$ in $Y$, is $A$ linear and continuous? My attempt: $A$ ...
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1answer
29 views

Exercise about an operator (adjoint and spectrum)

Let $y\in c_0$ and define the operator from $l^2 \rightarrow l^2$ as the following $$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$$ I have shown that the operator is continuous, compact and ...
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18 views

continuity of derivative of a differentiable function

Give an example of a function which is differentiable on [a,b] but it's derivative is not continuous on that interval. I already know one: F(x)= {x2.sin(1÷x),x is not equal to zero & x when x=0} ...
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44 views

Spectrum of self-adjoint operator on Hilbert space real

My book says that a self-adjoint bounded linear operator $A:H\to H$ on a complex Hilbert (not sure if separability is needed) space has a real spectrum. I guess that the key is in the fact that any ...
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26 views

$d$ is a metric space on $X\not=\{0\}$, obtained from a norm. $d'(x,y)=d(x,y)+1$. Show $d'$ cannot be obtained from a norm.

If $d$ is a metric on a vector space $X\not=\{0\}$ which is obtained from a norm, and $d'$ is defined by $d'(x,x)=0$, $d'(x,y)=d(x,y)+1, (x\not=y)$, show that $d'$ cannot be obtained from a norm. ...
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28 views

Finite dimensional spaces and R^n

I have a couple of questions, any assistance would be appreciated. I know that it can be shown that any finite dimensional space $M$ of dimension $N < \infty$ endowed with an inner product can be ...
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74 views

sums of projections in C*-algebras

Let $A$ be a $C$*-algebra with unit $e$. If $p$ and $q$ are projections such that $p+q+\lambda e$ is a projection for some $\lambda\in\mathbb{R}$, is it true that the only possible values for ...
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41 views

Containment of $c_0$ or $\ell_p$

Suppose that $(x_n)$ is a sequence of unit vectors in a Banach space $X$ such that $$\mbox{dist}(x_m, S_{X_n})=1$$ for all $m > n$. Here $S_{X_n}$ stands for the unit sphere $\mbox{span}\{x_1, ...
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16 views

Fundamental Lemma of CdV proof

Do you know an easy proof of the following result? Let $f\in L^1_{loc}(U)$, where $U$ is an open subset of the $n$-dimensional euclidean space. If $$\int_U f\varphi ...
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43 views

Spectrum of Verschiebung

I read that the shift operator $A:\ell_2\to\ell_2$, $(x_1,x_2,x_3,...)\mapsto(0,x_1,x_2,...)$ contains $0$ in its spectrum, and that's clear to me. It is also clear to me that it has no eigenvalue. ...
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33 views

Bounded linear operator commuting with every compact operators

Let $A$ be a bounded linear operator on the Banach space $X$. Assuming that $AK = KA$ for every compact operator $K$, how do I show that $A$ must be a scalar multiple of the identity, i.e., we have $A ...
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19 views

Existence theorem in Gilbarg and Trudinger

When attending talks in PDE I often heard "existence proof follow from Gilbarg and Trudinger..." Could anyone tell me rough what is the existence theorem for elliptic PDE roughly about? (My ...
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40 views

Invertibility of an operator involving inner product

Let $H$ be a Hilbert space with basis $b_i$. For all $t$, let $f(t;\cdot,\cdot)$ be an inner product on $H$. For each $j$, is $$\int_0^T \sum_{i=1}^\infty f(t,b_i,b_j)x_j(t)=0$$ uniquely solvable for ...
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22 views

Bounded resolvent

I am finding a somewhat obscure wording on my functional analysis text. $A$ is a linear operator on a complex Banach space $E$, which I'm almost sure to mean that $A$ is $E\to E$. Then I read that if ...
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2answers
34 views

Is $f(x) \in L^p(\mathbb R)$ always bounded for $x\longrightarrow\pm\infty$?

I need to prove the following result on the derivative of an Hilbert transform for $f,f'\in L^p(\mathbb R)$ $$\mathcal H\bigg\{\frac{df(x)}{dx}\bigg\}=\frac{d}{dx}\mathcal Hf(x) $$ In particular ...
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Continuous Quadratic Form $\implies$ Continuous Sesquilinear Form

Given a Hilbert space $\mathcal{H}$. Consider a quadratic form $q:\mathcal{H}\to\mathbb{C}$. Define its inducing sesquilinear form: $$s:\mathcal{H}\times\mathcal{H}\to\mathbb{C}: ...
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42 views

Are all definite integrals considered functionals?

In my Optimization class, we are messing around with some Calculus of Variations in an effort to find functions which minimize functionals. In these cases, the spaces we're working with are spaces ...
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boundedness of ball-like convex subsets in $\mathbb{R}^n$

Let $K \subset \mathbb{R}^n$ be a subset with the following three properties: (i) $K$ is convex (ii) $K$ is symmetric about $0$, that is if $k \in K$, then $-k \in K$ as well. (iii) If $l$ is any ...
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Lecture notes on semi group theory for linear evolution equations

I am reading (or trying to read :)) One parameter semigroups for Linear Evolution equations by Klaus-Jochen Engel and Rainer Nagel. I was wondering if someone was aware of a good set of lecture notes ...
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20 views

Banach Algebras: Continuity of Inversion?

Context: This question is related to this thread: Spaces of Functions Given a topological space $X$ and a Banach algebra with unit $B$. Consider a continuous map $F:X\to B$ that is invertible ...
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23 views

Is $\sum c_n z^n$ analytic when $c_n$ is Banach-valued?

I'm trying to define "Analytic function". I want a definition that covers all interesting cases. To be specific, let me explain what exactly I want Here is the definition of analytic function in ...
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1answer
17 views

Infinite-dimensional version of Gram matrix is invertible

We all know that a Gram matrix (a matrix with entries that are inner products of basis functions) is a invertible. Suppose I have $a_{ij} = (h_i, h_j)_H$ where the $h_j$ are basis functions of a ...
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1answer
42 views

For which exponents $\gamma$, the function $|x|^{1/2}$ is $\gamma$-Holder continuous?

I have to prove the following. Let $$u(x):=|x|^{1/2}$$ if $$|x|\le 1$$ For which exponents $\gamma\in (0,1]$, $u\in C^{0,\gamma}([-1,1]).$ The answer should be $\gamma\in (0,\frac{1}{2}]$, but I ...
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37 views

Isomorphism between Euclidean space and its conjugate

I know that, if $H$ is a Hilbert space, for any continuous linear functional $f\in H^{\ast}$ there is a unique element $x_0\in H$ such that $\forall x\in H\quad f(x)=\langle x,x_0\rangle$. Moreover, ...
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Local and global minimum/maximum and continuity

My task was to determine the global minimum of a function $f(x,y) = x^3 + y^3 - 3xy$ on the square $[0,2] \times [0,2]$. I first calculated the points where the gradient $(grad f)(x,y) = (3x^2 - 3y, ...
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$\phi_n \rightarrow \phi$ weakly-$*$, then $\|\phi\|\leq \limsup_n \|\phi_n\|$.

$\phi_n \rightarrow \phi$ weakly-$*$, then $\|\phi\|\leq \limsup_n \|\phi_n\|$. My attempt: $$\|\phi\| = \sup_{\|x\| = 1} |\phi(x)| \leq \sup_{\|x\| = 1} \lim_n |\phi_n(x)|$$ Using an ...
2
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1answer
47 views

show that every compact operator on Banach space is a norm-limit of finite rank operators (in a particular way, under the given hypotheses)

Let $X$ be a Banach space and suppose there is a net $\{F_i\}$ of finite-rank operators on $X$ such that $(a)$ $\sup_i||F_i||<\infty$ ; $(b)$ $||F_ix-x||\to 0$ for all $x$ in $X$. Show that if $A$ ...
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25 views

Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...
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Reference request for the proof of the Brodskii–Milman fixed point theroem for isometries

Can any one help me to access the paper M.S Brodskii and D.P Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948) 837–840 in Russian? or to prove the theorem If $K$ is a ...
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20 views

Show that $\{x_1,…,x_n\}$, where $x_j=t^j$, is a linearly independent set in the space $C[a,b]$.

Show that $\{x_1,...,x_n\}$, where $x_j=t^j$, is a linearly independent set in the space $C[a,b]$. I think I can use properties of polynomials in $R[x]$ here, but I'm not sure. Using $\sum_{i=1}^n ...
3
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36 views

Existence theorem on weak solutions of ordinary differential equations

Consider an ordinary vector-valued differential equation of the form $$ \begin{align*} \dot y(t) &= f(t,y(t)), \\ y(0) &= y_0 \in \mathbb{R}^n. \end{align*} $$ It is well known that if $f$ is ...
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1answer
26 views

Dirac delta distribution & integration against locally integrable function

I was reading the a lecture note online about distribution theory and it said: The Dirac delta distribution $\delta \in D'$ is defined as $\delta(\varphi)= \varphi(0) $, and there's no locally ...
2
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27 views

Maximal chain in the collection of all invariant subspaces for compact operator $K$

Let $X$ be a Banach space over ${\Bbb C}$, and $K\in K(X)$ ($K(X) = $ compact operators space). Show that if ${\cal L}$ is a maximal chain in the $Lat K$ ($Lat K = $ the collection of all invariant ...
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1answer
51 views

Operators on non-separable Banach spaces have non-trivial invariant subspaces

Show that if $T\in B(X)$ and $X$ is not separable, then $T$ has a nontrivial invariant subspace. I know that $\ker (T)$ and $\operatorname{ran}(T)$ are invariant $T$-subspace. So if $\ker T\neq ...