# 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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### Fixed Point method and existence of entropy solution of a Nonlinear Parabolic PDE [on hold]

Could someone point me to literature that addresses this kind of problem? Actually I want to prove the existence of entropy solution for a nonlinear degenerate PDE for an equation that cannot solve ...
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### Sot convergence of a net

The following are exercises of Conway's operator theory: I proved both exercises, but I confused about this point that in exercise 8, $T_i\to 0$ (sot), so based on exercise 6, $T_i^2 = T_i.T_i\to 0$...
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### Closure of a set in the weak topology

Let $X$ be a Banach space, $S$ a subset of $X$. What is the closure of $S$ with respect to the weak topology?
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### Does Taylor series around point zero (maclaurin series) always exist for differentiable function?

For every differentiable function $f(x)$, is $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n!} \, (x)^{n}$ always true and can be written? The only thing we have to be concerned is just whether ...
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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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### What is the general form of linear operators on continuous functions?

I was wondering if there was a representation for a set of operators dense in the space of linear operators $B$ mapping $C(a,b) \to C(c,d)$. I thought that maybe integral operators give a general ...
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### Is a contraction idempotent operator self-adjoint?

Is a contraction idempotent operator self-adjoint? In the other words, if $T:H\to H$ is a bounded linear operator such that $||T||\leq1$ and $T^{2}=T$, can we conclude $T=T^*$?
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### Continuous family of subalgebras in a C* algebra

Let A be a separable C* algebra. Fix $n\in\mathbb N$. For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that: $A_t \cong \mathcal{O}_n$ (Cuntz algebra). Generators of $A_t$ depend ...
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### Differentiability in fractional Zygmund spaces

I'm trying to understand why for $s$ not an integer, if $$S_0u\in L^\infty\text{ and }\sup_{j\geq0}2^{js}\|\Delta_ju\|_{L^\infty}<\infty,$$ then $u\in C^s$. Here I write $\Delta_j$ for the ...
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### Dual norm intuition

The dual of a norm $\|\cdot \|$ is defined as: $$\|z\|_* = \sup \{ z^Tx \text{ } | \text{ } \|x\| \le 1\}$$ Could anybody give me an intuition of this concept? I know the definition, I am using it ...
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### How do you define the inverse of an (exponential Lie) operator?

I know this is a fairly general question, but I would like to know anything I can about obtaining the inverse of an exponential of a lie operator. More specifically, I want to know how one can ...
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### Addition of $X^*$ is continuous w.r.t. Weak Topology $\sigma(X^*, X)$

I showed that the addition on $X^*$ is continuous w.r.t. Weak Topology $\sigma(X^*, X)$. Since I'm a newbie in this field, would you please check my proof and answer that my proof is correct or not? ...
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### How tight is this trace inequality?

I would like to know how tight the following trace inequalities are for real symmetric $A$ and real symmetric $B \succeq 0$ $$\mbox{trace} (AB) \leq \lambda_{\max} (A) \cdot \mbox{trace} (B)$$ or ...
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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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### Generator of an analytic semigroup with a compact resolvent --> 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 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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### 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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### 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?
Exercise: For which $p\in [1,\infty]$ is $L^p([0,1])$ strictly convex? Solution: For strict convexity we have two equivalent definition: If $x\neq 0\neq y$ and $\Vert x+y\Vert=\Vert x\Vert+\Vert ... 1answer 19 views ###$C([0,1])$strictly convex My question is quite simple: Is the space$C([0,1])$strictly convex? Where strict convexity is defined as: if$\Vert y+x \Vert=\Vert y\Vert+\Vert x\Vert$then it implies that is exists an$\lambda&...
Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: \frac{d}{dt}\...