# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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.
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 ...
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 ...