# 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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### A necessary and sufficient condition such that product of partial isometries is a partial isometry

I'm reading the paper P. Halmos, L. Wallen, Powers of Partial Isometries, Indiana Univ. Math. J. 19 No. 8 (1970), 657–663 (http://www.iumj.indiana.edu/docs/19054/19054.asp). And I got stuck on the ...
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### Separation of closed convex sets in finite dimensional space

If $A, B \subset \mathbb R^n$ are closed, convex, and disjoint, is there a vector $a \in \mathbb R^n$ such that $a^t x < a^t y$ for all $x \in A$ and $y \in B$? I found many theorems requiring one ...
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### Creating a “nice” R^2->R function from partial subdomains

This question is probably poorly defined, mainly because I am still not entirely sure what I need, and I just need some ideas to start with. I have a R^2 domain, with a set of given curves ...
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### Compact set of probability measures

I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
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### Does a generator of an analytic semigroup with a compact resolvent has pairwise conjugate eigenvalues?

I am reviewing a paper in which the authors claim that their operator has eigenvalues $\lambda$ that are either real or pairwise conjugate (meaning that if $\lambda$ is an eigenvalue, then also the ...
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### A Question on Normal Operators

Let $T$ be a compact operator on a Hilbert space $H$. I want to prove the following: If there exists Orthonormal Basis from the eigen vectors of $T$, then $T$ is normal operator.
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### Involution and Gelfand Transform Properties

Let $\mathcal{B}$ be a commutative unital Banach algebra, and let for each $x\in\mathcal{B}$ $\hat{x}$ be the Gelfand transform. I assume that $\mathcal{B}$ has an involution *. I want to show that: ...
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### a proof that $c_0$ is a Banach space

I'm currently reading the functional analysis lecture notes taught in MIT, and I came across a filling-in on the part of the reader. I wonder if I've filled in the details as expected by the author. I ...
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### Dual of the Banach space of $k$-times continuously differentiable functions.

Let $C^k([0,1])$ denote the Banach space of $k$-times continuously differentiable functions $f:[0,1]\to \mathbb R$ with norm $$\|f\|_{C^k}:=\max_{i=0,\dots,k}\sup_{x\in [0,1]}|f^{i}(x)|.$$ I'm trying ...
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### Lower bound on gap between consecutive eigenvalues on $L_2(\mathbb{R}^3)$

A similar version of this question was originally posted by me in the physics community, but it was suggested that I ask the mathematicians instead. So I have tried to strip off most of the physics ...
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### If a map between separable Banach spaces has closed graph, does it have a point of continuity?

It is well known that the closed graph theorem does not directly extend to nonlinear maps: even for functions from $\mathbb{R}$ to $\mathbb{R}$, having closed graph does not imply continuity. But let'...
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### Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
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### Why is uniqueness important for PDEs?

Every text on PDEs I come across will spend alot of time on showing the existence and uniqueness of solutions to a particular PDE. The importance of the existence of a solution to a PDE is obvious, ...
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### The Set of Extremal Point Need not be closed

Exercise: Consider the Convex Hull $C$ of the points $(0,0,\pm 1)$ and the circle $\{(1+\cos\varphi,\sin\varphi,0): 0\leq \varphi \leq 2\pi\}$ in $\mathbb{R}^3$. Determine the extremal point of $C$. ...
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### Radon probability measures on $X$: Determine the Weak$^{\ast}$ closure

Exercise: Equip the set $P(x)=\{\mu \in C_0(X,\mathbb{R})^{\ast}: \mu\geq 0, \Vert \mu\Vert=1\}$ with the weak$^{\ast}$-topology. There is a map $\delta:X\to P(X)$, $x\mapsto \delta_x$ given by: \...
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### is this an counterexample for: $(C[a,b],\| \cdot \|_2)$ is complete?

our prof wanted to show that $(C[0,1],\| \cdot \|_2)$ is not complete. So he said $$f_k(x) = x^k$$ is a counterexample. I wonder if this is true. I tried to show that $f_k$ is cauchy sequence. But i ...
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### Space of continuous functions with compact support dense in space of continuous functions vanishing at infinity

How can we prove that the space of continuous functions with compact support is dense in the space of continuous functions that vanish at infinity?
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### Extreme points of the unit ball of the space $c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$

I want to prove that all "closed unit ball" of $$c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$$ do not have any extreme point. Would you please help me? (Extreme ...
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### application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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### Compact operator problem: $I-T$ is onto, then $I-T$ is invertible?

I want to show the following: Let $T$ be a compact operator in a Hilbert space $H$. If $I-T$ is onto, then $I-T$ is invertible. Would you show me how to prove this argument? Or please tell me some ...
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