# 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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### Convolution operator is normal

Consider the convolution operator $$Tf(s)=\frac{1}{2\pi}\int_0^{2\pi}f(t)h(s-t)\,\,dt,\quad f\in L^2[0,2\pi]$$ where $h:\Bbb R\to \Bbb C$ is a $2\pi$-periodic function, square integrable on ...
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### Can one prove the existence of a fixed point for a shrinking map on a sequentially compact metric space WITHOUT proving the space is compact?

Let $(X,\rho)$ be a metric space with $Y\subset X$ a sequentially compact subspace, and a mapping $T:X\to Y$ satisfying $\rho(Tx, Ty)<\rho(x,y)$ for all $x\neq y$. Prove that $T$ has a unique ...
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### banach space bigger than $L^p$

we know that $L^p$ is banach space for any $p\geq 1$. My question: Is there any other banach space that is bigger than $L^p$?. In fact, I have an exercice that I don't have any idea: prove that ...
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### What does $u_0(x)$ represent?

I am looking at the heat equation and in my notes it says the initial temperature distribution $u(0,x)=u_0(x)$. what does this mean? What does $u_0(x)$ represent?
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### Fermat like equation for meromorphic functions.

I found this question in Conway, and really have no idea how to answer it. Can anyone provide any hints? For each integer $n\geq 1$ determine all meromorphic functions on $\mathbb{C}$ $f$ and $g$ ...
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### Where does $u(t,x) \to u(t,x)-a-(b-a)x$ come from?

I know the heat equation is $$\frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)$$ I know that $u(t,x)$ is the temperature distribution at time $t$ at the point $x$. We assume ...
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### Weak uniform convergence

Let $(X,\|\cdot\|)$ a reflexive and separable Banach space, and note by $X^{*}$ its topological dual and $\omega$ its weak topology. Also, put $C_{\omega}(I,X)$ the space of the continuous mappings ...
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### Trouble understand a step in the proof that $l^p$ is complete

I'm reading through a proof, attached here. I didn't include the whole proof. The last step is the one I'm confused about. Shouldn't there be more of a justication for taking $\lim_{n \to \infty}$ ...
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### Tools to study level sets of Lebesgue function

Given $u\in L^\infty(\mathbb{R}^N)$ with compact support, are there nice tools to study the level sets $$K: = \{x\in \mathbb{R}^N : u(x) = \|u\|_\infty \text{ a.e.}\}.$$ The main problem is $K$ being ...
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### Reference request: about inverse Laplacian operator

I am currently studying some problems about inverse Laplacian and the Yosida approximation and wishing to learn more about it. Here is a post about one of the problems that I am interested in. ...
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### Product spaces and open sets

I have a proposition I have been pondering that I need help with. Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. Recall that the product space $(X\times Y, d_{1})$ is also a metric space with the ...
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### Commuting nets for commuting projections

Let $A$ be a $C$*-algebra and $p,q\in A^{**}$ be commuting projections. Then there exist self-adjoint nets $(x_i)_i$ and $(y_j)_j$ in $A$ with $x_i\to p$ and $y_j\to q$ in the weak *-topology. Can ...
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### Continuous but not compact operator on $L^2(0,\infty)$

Define the following operator on $L^2(0,\infty)$: $$Tf(x)=\frac{1}{x} \int_0^xf(y)dy,\quad f\in L^2(0\infty).$$ I would like to see that it is continuous but not compact. So, this is an integral ...
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### Why the set of pure state ‎is ‎weak* ‎compact?

Let ‎$‎‎A$ ‎be a‎ ‎C*-algebra‎. ‎ ‎$‎S(A)‎$ ‎is ‎the ‎set ‎of ‎state ‎on ‎‎$‎‎A$ and $‎‎PS(A)$ ‎is ‎the ‎set ‎of ‎pure ‎state ‎on ‎‎$‎‎A$. ‎ ‎ I ‎know ‎that ‎if ‎‎$‎‎A$ ‎is ‎unital ‎then ‎‎$‎‎S(A)$ ...
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### Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...
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### The definition of metric space,topological space

I have read some books in analysis. All of them define metric space, topological space or vector space directly, without any reason. Therefore, I want to know the background of the definition - the ...
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### Arzela-Ascoli for $\mathbb R^n$ from the case of $\mathbb R$?

In class, we proved the Arzela-Ascoli theorem for $\mathbb R$. The lecturer said it's also true for $\mathbb R^n$, and this version is deducible from $\mathbb R$. I tried to do this but failed. How ...
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### Prove that $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent

Consider A to be a matrix that has all eigenvalues $\lambda$ with negative real part, that is, $Re(\lambda) < 0$. a)Show that the integral $\int_{0}^\infty (e^{As})^T e^{As}\,ds$ is convergent, ...
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### Exists a uniformly convex norm on Banach space satisfying certain condition?

Let $E$ be a Banach space with norm $\|\cdot\|$. Assume that there exists on $E$ an equivalent norm, denoted by $|\cdot|$, that is uniformly convex. Given any $k > 1$, does there exist a uniformly ...
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### Positive self-adjoint operators and norm resolvent convergence

I recently came across a reference to the following Theorem (Simon/Reed, Methods of Modern Mathematical Physics, viii.25) and am now trying to figure out a proof for it: If $A_n$ and $A$ are positive ...
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### Properties of mollification

We have this theorem For any $1\le p<\infty$ and $f\in L^p(\mathbb{R}^k)$, then $\|f*\phi_\delta - f\|_p\to 0$ as $\delta\to0$, where $\phi$ is any nonnegative measurable function on ...
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### Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$T = T^* ~\text{but}~ \sigma_p(T)= \emptyset$$ Some definitions and results of the ...
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