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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30 views

Fourier transformed multiplication operator leaves $L^2([-C,C])$ invariant?

Let $C > 0$ be some constant and $L^2([-C,C])$ the square integrable functions on $[-C,C]$. Let $\delta > 0$ and let $M_{|\cdot |^\delta}$ denote the multiplication operator on $L^2(\mathbb R)$ ...
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27 views

differential of the regular distribution in the space $D'$

Determine the differential of the regular distribution $T_f$ in the space $D'$(continuous dual of $D$) for $f(x)=H(x)cos(x)$, where $H$ is a Heaviside function and $x\in \Bbb{R}$. Since $H(x) = +1$ ...
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46 views

Formula for differential of $\exp$ at a Banach algebra.

In Rossman (Lie Groups - An introduction through linear groups), he makes the following statement: Theorem: $$\exp'_X(Y)=\exp(X)\frac{1-\exp(-ad_X)}{ad_X} Y,$$ where ...
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54 views

Real Analysis, Folland problem 5.5.60 Hilbert Spaces

problem 5.5.60 - Let $(X,M,\mu)$ be a measure space. If $E\in M$, we identify $L^2(E,\mu)$ with the subspace of $L^2(X,\mu)$ consisting of functions that vanish outside $E$. If $\{E_n\}$ is a ...
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53 views

Does this sequence of polynomials converge to the square root function?

Taken from Lang's R & F Analysis (p.60). For some reason I can't see why, for $t \in [0,1]$, the following is true for all natural numbers $n$ (by an inductive argument): $$0 \leq \sqrt{t} - ...
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83 views

How do I show this is a solution for this differential equation?

Consider that $$E(t,x)=\dfrac{H(t)}{2\sqrt{\pi t}}e^{-|x|^2/4t}.$$ I want to show that $E_t - E_{xx} = \delta(t)\delta(x)$. This means that we need to show that if $\phi\in ...
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30 views

What is the error in this proof about compact operators?

Assume $C:U\to V$ is compact and $\|M_nx-Mx\|\to 0$ as $n\to\infty$ for all $x\in X$. $\begin{aligned} \|CM_n-CM\|&=\sup_{\|x\|=1}\|(CM_n-CM)x\|\\ &\leq\sup_{\|x\|=1}\|C\|\|M_nx-Mx\|\\ ...
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71 views

Results about Hilbert-Sobolev space with homogeneous boundary condition.

I am currently reading works about numerical method for solving differential equations. The main setting of the work revolves on the space $H^m_0[0,1]$, which is defined by $$ H^m_0[0,1]:=\{f\in ...
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24 views

If $f \in L^{\infty}(\Bbb R^d) \cap C_u(\Omega)$, then $\rho_{\epsilon} * f \to f$ uniformly over $\Omega$

Consider the following statement: If $(\rho_{\epsilon})_{\epsilon > 0}$ is an approximation identity and $f \in L^{\infty}(\Bbb R^d) \cap C_u(\Omega)$, then $\rho_{\epsilon} * f \to f$ ...
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30 views

Exercise on Hilbert spaces and complete orthonormal systems

Let $H$ be a Hilbert space of finite dimension $N$. Prove that every complete orthonormal system in $H$ has $N$ elements and that $H$ is linearly isometric to $\mathbb{R}^N$. I can't start with this ...
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22 views

A naïvely constructed extrapolation of a self-adjoint operator. Is it self-adjoint?

Let $\mathcal{H}$ be a real Hilbert space and let $A\colon D(A)\subset \mathcal{H}\to \mathcal{H}$ be an unbounded operator. Consider also a Hilbert triple $$ \mathcal{H}_+\subset \mathcal{H}\subset ...
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22 views

When does a Lie bracket exist for a Frechet manifold?

Does a general Frechet manifold admit a Lie bracket? A bracket can certainly be constructed in certain cases, but my guess is that it is wrong to assume that one exists in general.
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23 views

Prove $(\Bbb R^{n},d_{p})$ forms a complete metric space

Prove $(\Bbb R^{n},d_{p})$ forms a complete metric space So I know that in order for a metric space to be complete I must show that every Cauchy sequence in $\Bbb R^{n}$ converges on $\Bbb R^{n}$ ...
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22 views

Cauchy Schwarz inequality on scalar terms

The following equations are derived from Quantum information theory, which requires the use of Cauchy Schwarz inequality for a proof. I am quite puzzled by the second term in the summation, which ...
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19 views

Showing convergence of sequence of functions?

Let $(z_n )_n∈N$ be a sequence of non-zero complex numbers such that $|z_n |$ → ∞ as n → ∞. Let r > 0. Show that the sequence of functions $\prod_{k=0}^n E_k (\frac{z}{zk} )$ converges uniformly on ...
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43 views

Kuratowski Embedding Theorem

I have a quesion to the following proof of the Kuratowski Embedding theorem on page 37/38. ...
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14 views

Representation of Weyl algebra

Let's consider an algebra $W$, generated by a family of differential operators of type $$\sum_{k=0}^{n}{a_{k}(x) \cdot \frac{d^{k}}{dx^{k}}}$$ (may also known as Weyl algebra). I would like to prove ...
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40 views

Is this how we define “limit in the distributional sense”?

Consider $\mathcal{D}(\mathbb{R})$ the space of test functions and $\mathcal{D}'(\mathbb{R})$ the space of distributions, in $\mathbb{R}$, i.e., continuous linear functionals over ...
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36 views

If $B = \left\{ {z \in \mathbb{C}:{s_n}(z) = {s_{n-1}}(z)} \right\}$ then $\operatorname{interior}(B)=\phi$

let $A_j \in \mathbb{C}^{n \times n}$ $j = 0,1,2,\ldots,m$ $P(z) = A_m z^m + \cdots + A_1 z + A_0$ is matrix polynomial and $z$ is a complex variable. $s_1 \ge s_2 \ge \cdots \ge s_n$ are singular ...
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20 views

Let $F = \left\{ {({A_1} + {B_1})x + ({A_2} + {B_2}):\left\| {{B_i}} \right\| \le {\alpha _i},i = 1,2} \right\}$.What is boundary of $F$?

Let $F = \left\{ {({A_1} + {B_1})x + ({A_2} + {B_2}):\left\| {{B_i}} \right\| \le {\alpha _i},i = 1,2} \right\}$ such that, $A_i,B_i\in M_n$ and $\alpha_i>0$ for all $i$ and $x$ is a complex ...
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30 views

A question on connected component

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0\le w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$, and $w_o>0$. ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + A_0$ is a matrix polynomial, where $x ...
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25 views

About a sequence of functions that converges locally but not globally

Good morning. During my thesis, I have come to the following problem: suppose $(M, g)$ is a closed Riemannian manifold of dimensione greater than $2$. You have a function $\varphi \in C^0(M)$ s.t. ...
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33 views

Description of Frechet derivatives and the implicit function theorem

[QUESTION] Let $(S^n,\bar{g})$ be the unit sphere and $h$ be another Riemannian metric on $S^n$, $0<\alpha<1$. $M^{2+\alpha}(S^n):=\left\{F:S^n\stackrel{C^{2,\alpha}}\to S^n\right\}$. For ...
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115 views

Taylor's series question on bounds

$f(x) = f(y) + f'(y)(x-y) + \frac{f"(y) (x-y)^2}{2} + ....$ is the taylor's series. I am aware of bounds which specify the error in approximating $f$ with a polynomial. I want sufficient conditions ...
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44 views

Spectrum of the derivative operator: What's wrong in my argument?

Consider the Banach space $X=C[0,1]$ of continuous functions $f:[0,1]\to\mathbb{R}$ equipped with the supremum norm. If we consider the following unbounded operator $A$ defined on its domain ...
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40 views

What is the difference between $H^1_{loc}$ and $H^1$?

I have started studying Sobolev spaces and I came across a space referred to as $H^1_{loc}$. I am not sure what the $loc$ subscript infers? What is it that makes this space different from $H^1$? Why ...
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34 views

Is every Boyd-Wong mapping also a contraction?

I know every contraction is a BW map, so I suspect the BW class of mappings is strictly larger. Can someone help me construct an example of a mapping that is not a contraction but is a BW map? Edit: ...
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52 views

irrotational vector-field => Existence of scalar potential - for Sobolevfunctions

For $v\in C^1(\Omega,\mathbb{R}^n)$ the following is well-known: Let $\Omega \subset \mathbb{R}^n$ be simply connected. Then for every $v\in C^1(\Omega,\mathbb{R}^n)$ with $curl \;v = 0$ there exists ...
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16 views

Conditions for weak differentiability of composition of $C^1$ real function with weakly time-differentiable $H^1$-valued function

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and $\mathbb{R} \ni T > 0$. I will abbreviate $X=H^1(\Omega)$ and write $X'$ for its topological dual. Given $$u\in L^2\left(0,T;X \right)$$ ...
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23 views

Does every tensor norm satisfy one half of the crossnorm property

Suppose $(H,\|\cdot\|_{H})$ and $(G,\|\cdot\|_{G})$ are normed vector spaces. Suppose $\|\cdot\|$ is a norm on their algebraic tensor space $H\otimes G$. Do we have $$ \|h\otimes g\|\leq ...
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10 views

Closed sublattice generated by countable set

Apparently the closed sublattice generated by a countable subset of a Banach lattice is separable. I am trying proof this, but I am stuck, does anybody have an idea? Trying to prove the above, I ...
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38 views

Integral operator and eigenvalues

Let $T:L^2[0,1]\to L^2[0,1]$ be a compact self-adjoint operator, and let $(\lambda_n)$ be the non-zero eigenvalues of $T$ a) Show that $T$ is the integral operator associated to some $k\in ...
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22 views

Approximating step functions by Haar wavelets

Let $\psi = \chi_{[0,1/2)} - \chi_{[1/2,1)}$, then $\psi_{n,k}(t) = 2^{n/2}\psi(2^nt-k)$ with $n \in \mathbb{N}$ and $k \in \{0,1,\dots,2^n-1\}$ defines the Haar-Wavelets on $L^2(0,1)$. Let $S$ be the ...
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21 views

System of linear Volterra integral equation

Consider the Volterra integral equation $$ f(t) = g(t) +\int_0^t K(t,s) f(s) ds $$ where $g(t)$ is continuous on $t\in[0,T]$ and $K(t,s)$ is a weakly singular kernel. It is well known that there ...
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46 views

Let $A = \left\{ P_\Delta (\lambda ):\| \Delta_j \| \le \varepsilon w_j ,j = 0,1,2,\ldots,m \right\}$.What is boundary of $A$?

Suppose $P_\Delta(\lambda ) = (A_m + \Delta_m) \lambda ^m + \cdots + (A_1 + \Delta _1)\lambda ^1 + (A_0 + \Delta _0)$ is a matrix polynomial, and $\lambda $ is a complex variable. $A_j,\Delta _j ...
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33 views

Usefulness of absolute value in optimization algorithms

In a course of Optimization Algorithms at university, professor said that in every algorithm the objective/object function/function cost is defined as: $$f(\bar x)=\lvert x_0 - g(\bar x)\rvert^{2}$$ ...
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30 views

Applying a general function an infinite number of times

I am trying to learn more about infinite application of functions and functionals. My background is in quantum chemistry, so please forgive some of my notation and terminology. Motivation In ...
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26 views

Help to verify (numerically) invariant Haar measure on unitary group

This question is related tot he paper http://gemma.ujf.cas.cz/~brauner/files/Haar_measure.pdf I am interested to understand and verify equation (3). Can anyone please help? My present state: As I ...
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Extreme points of the closed unit sphere in $\big(C_K([0,1]),\|\;\|_\infty\big)$

Lets take $(C_K([0,1]),\|\|_\infty)$, which is a normed vector space. $$ C_K([0,1])=\{x:[0,1]\to K\;\;|\;x\text{ is continuous } \}\\ \|x\|_\infty=\sup_{t\in[0,1]}|x(t)| $$ So, since the closed unit ...
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40 views

orthonormal basis in $L^2$ space

Let $\{\phi_i (x)\}_{i=1}^\infty$ be an orthonormal basis for $L^2 (S)$. Prove that $\{\psi_{ij} (x,y) = \phi_i (x) \phi_j (y)\}_{i,j=1}^\infty $ is an orthonormal basis for $L^2 (S \times S)$. Thanks ...
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37 views

Is this a usual approach to uniform convergence?

Consider $E\subset \mathbb{R}$ and a sequence of functions $(f_n)_{n\in \mathbb{N}}$ with $f_n : E\to \mathbb{R}$. The easiest form of convergence we can define is the pointwise convergence - we say ...
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10 views

Two characterizations for the interior and closure of a convex and absorbing set when the Minkowski functional is continuous

Let $(X,\|\;\|)$ be a normed vector space over $K$ and E$\subset X$ be convex and absorbing. Lets define $E_1=\{x\in X:p_E(x)<1\}$ and $E_2=\{x\in X: p_E(x)\le 1\}$. I want to prove that: $$ ...
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34 views

Why is $(\sqrt{P})^2=P$ where $P$ is a positive operator on a Hilbert space?

The following is a proposition regarding positive operators on a Hilbert space in Douglas's Banach Algebra Techniques in Operator Theory: Corollary 4.32 is as the following: I understand that the ...
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34 views

Theorem (Continuous compact function in compact $E\subset\mathbb{R}^n$)

Above all, I want to say I am Sorry for posting many questions. Theorem 1.15 (Continuous compact function in compact $E\subset\mathbb{R}^n$) Let $E$ be a compact set in $\mathbb{R}^n$ and $f$ be ...
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26 views

Prove generalisation of Sobolev embedding theorem using induction.

I am trying to prove the following; I am doing it by induction and the case $k=1$ is already done. So suppose the above is true for all integers less than or equal to k, first we want to show that ...
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30 views

approximate $C^1$ function by Holder function

I am trying to prove the following Lemma: Suppose $u$: $\mathcal R^N\to \mathbb R$ is $C^1$. Then for each $\epsilon>0$, there exists a $C^{1,\alpha}$ function $\tilde f$ such that $$ ...
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51 views

What is different between $\frac{1}{1+\lambda x}$ and $\exp{(-\lambda x)}$

I want to choose a function $f(x)$ which has properties: $f(x)$ closes to $0$ when $x$ goes to $+\infty$ . I have two option for that $f(x)=\frac{1}{1+\lambda x}$, where $\lambda$ is tuning ...
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18 views

If F is convex and lower semicontinuous in norm, then F is weakly lower semicontinuous

I have to prove the following statement: If $F$ is convex and lower semicontinuous in norm, then $F$ is weakly lower semicontinuous.
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50 views

Do compact convergence topology and w*-topology coincide on the Pontryagin dual group of a LCA group.

Given a locally compact group $G$, consider its Pontryagin dual $G^{'}$. Do compact convergence topology and weak*-topology on $G^{'}$ coincide?. When we talk about weak*-topology on $G^{'}$, it ...
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45 views

Products of functions is Hölder continous

I have $A$ maps $\Omega$ into the $n\times n$ real matrices, where $\Omega$ is a open bounded subset of $\mathbb{R}^N$. $C^{\alpha}(\Omega)$ denotes the set of functions that are Hölder continuous ...