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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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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26 views

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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10 views

Domain Definition for Gradient Operator

I want to define to gradient operator $\nabla f$ on an $n-$variable function $f\left(x_1,\cdots,x_n\right)$. For this purpose I want to well define the spaces that the domain and the range of the ...
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1answer
25 views

Difference between little o and big O notation in taylor expansion

I know I can say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+o(\Delta t) $$ But can I say that: $$ y(t+\Delta t)=y(t)+y'(t)\Delta t+O((\Delta t)^2) $$ In both cases, do I have to add that $\Delta t \...
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1answer
29 views

Is there a name for the “scalar product” in Banach spaces?

What would be the best term for the same object i.e $(x,y)$. I guess its some kind of bilinear form, but in general it does not obey the parallellogram law hence its not a scalar- or innerproduct. ...
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40 views

“Of the order of” notation

I have a function where all terms have the same coefficient $x^3$ in it: For example $f(x) = ax^3 - bx^3$ Can I say that in big $O$ notation: $f(x) = O(x^3)$ $f(x) = O(x^3)$ as $x \rightarrow 0$ ...
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1answer
22 views

Show space of C1 functions on (0,1) is a Banach lattice

I'm working on the beginning of a book on Banach lattices, and it wants me to show that $C^1(0,1)$ is a Banach Lattice, with the norm $\|f\|=\|f'\|_\infty+|f(0)|$ and the order $f\le g$ if $f(0)\le f(...
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1answer
29 views

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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1answer
16 views

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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1answer
22 views

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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1answer
16 views

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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18 views

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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29 views

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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1answer
26 views

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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0answers
27 views

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 ...
2
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1answer
30 views

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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35 views

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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27 views

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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40 views

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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1answer
43 views

Is $C[a,b]$ isomorphic to $C([a,b]\cup[c,d])$

It is very simple to prove that $C[a,b]$ is isomorphic to subspace of $C([a,b]\cup[c,d])$ and vice versa $C([a,b]\cup[c,d])$ is isomorphic to subspace of $C[a,b]$. Is it true for $\bigl(C[a,b], C([a,...
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1answer
19 views

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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10 views

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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1answer
37 views

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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18 views

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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9 views

$L^p([0,1])$ stricly convex [duplicate]

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 ...
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15 views

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? Thanks
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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&...
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1answer
43 views

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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1answer
44 views

A Spectrum of a compact operator in $\ell^p$

Let $\alpha_n \in \mathbb C$ and $\lim_{n\to\infty}\alpha_n = 0$. Let $T$ be a linear continuous operator from $\ell^p \to \ell^p (1\le p\le \infty)$ defined by $$ T((x_1, x_2, \ldots)) = (\alpha_1 ...
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0answers
19 views

uniform convergence of lagrange polynomials , exercise 12.16.15 dieudonne treatise vol 2

This is exercise 12.16.15 from Dieudonne's treatise on analysis volume 2 It attempts to find necessary and sufficient conditions on a sequence of control points in I = [0,1] for the lagarange ...
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1answer
33 views

What is the operator norm of $(c_1 x_1, c_2, x_2, \cdots)$ in $\ell^p$?

I'm wondering that the operator norm $\|T\|$ where $T:\ell^p \to \ell^p$, $1\le p\le \infty$ is $$ Tx = (c_1 x_1, c_2x_2, \cdots), $$ where $c_k \in \mathbb C$ such that $\lim_{k\to\infty} c_k = 0$. ...
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1answer
45 views

Compact operator with closed image

Let $K$ be a compact operator between two normed spaces. If $K(X)$ is closed, does this necessarily imply that $K(B)$ is closed? where $B$ is the closed unit ball?
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1answer
19 views

Exercise: The range of Compact Operators

Exercise: Suppose $K:X\to Y$ is compact operator. Show that $K(X)\subseteq Y$ is separable Assume $Y$ is a separable Banach space. Find a Banach space $Z$ and a compact operator $K:Z\to Y$ s.t. $K(...
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1answer
26 views

A problem concerning compact operators in $\ell^p$ [duplicate]

Let $\alpha_n \in \mathbb C$ and $\lim_{n\to\infty}\alpha_n = 0$. Let $T$ be a linear continuous operator from $\ell^p \to \ell^p (1\le p\le \infty)$ defined by $$ T((x_1, x_2, \ldots)) = (\alpha_1 ...
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0answers
23 views

I don't see why $W^{1, 2}(\partial D)$ being compactly embedded in $L^2(\partial D)$ lets us show an operator is Fredholm of index zero.

Let $D$ be a bounded Lipschitz domain. Let $A$ be the single layer potential which maps $L^2(\partial D)$ into $W^{1, 2}(\partial D)$ boundedly. $A$ is given by: $$ A_D[\phi] = \int_{\partial D}G(x-y)...
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1answer
41 views

A simple $\ell^1$ norm question

Let $B$ be a closed subspace of $\ell^1$ such that $$ B=\{\{x_n\}_{n=1}^\infty \in \ell^1 : \sum_{n=1}^\infty \frac{n}{n+1} x_n = 0\} . $$ I want to show that there is NO $x = (x_1, x_2, \cdots) \in B$...
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0answers
30 views

The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
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33 views

An Extreme Point of a closed ball of $\ell^\infty$ [duplicate]

I am trying 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. (Extreme Point) Let $X$ be a ...
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2answers
33 views

$Im(K)\subset Y$ closed, infinite-dimensional: $K:X\to Y$ is not a compact operator

Proposition: If $K:X\to Y$ is a bounded linear operator between two Banach spaces $X$ and $Y$ such that $\operatorname{Im}(K)\subset Y$ is an infinite-dimensional closed subspace, then $K$ is NOT ...
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1answer
49 views

Space of $f(0)=f(1)$: Is Hilbert space?

Let $S$ be space consisting of collection of square integrable continuous functions $f:[0,1]\rightarrow\mathbb{R}$ with the constraint $f(0)=f(1)$. So $S$ is an inner product space with the inner ...
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15 views

Exercise involving sequence Palais-Smale and operator compact.

Let H be a Hilbert space and let K: H $\rightarrow{R}$ be $C^1$ and such that $\nabla K: H\rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H\rightarrow R$ defined by ...
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17 views

How to prove $\limsup\limits_{n\to\infty}\rho_k(x_n+x)=\limsup\limits_{n\to\infty}\rho_k(x_n)+\rho(x)?$ on $\ell_1$

Let $p(.)$ be an equivalent norm to the usual norm on $\ell_1$ such that $$\limsup\limits_{n\to\infty} p(x_n+x)=\limsup\limits_{n\to\infty}p(x_n)+p(x)$$ for every $w^*-$null sequence $(x_n)$ and for ...
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1answer
26 views

Operator linear and continuous, show that it is compact

Exercise: Let $X$, $Y$ be normed spaces and $A:X\to Y$ be a linear and continuous operator, which has the property \begin{equation} \exists C>0 \text{ s.t. } \forall x\in X \quad \Vert Ax\Vert\geq ...
2
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1answer
29 views

Why isn't $\ell^p$ locally convex for $0<p<1$?

I believe we have to distinguish the finite-dimensional from the infinite dimensional case. Regardless, if $0<p<1$, $\|x\|_p := (\sum |x_i|^p)^{\frac 1 p}$ is not a norm as it fails to satisfy ...
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3answers
31 views

Continuous linear operator that is NOT compact.

Exercise: Let $U:L^1(0,\infty)\to L^{\infty}(0,\infty)$, defined as \begin{equation}U(f)(x):=\int_0^x f(t)dt\end{equation} Prove that $U$ is linear, continuous but not compact. My solutions (or what ...
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0answers
9 views

Inclusions of Multiplier algebras associated to hereditary subalgebras

I have been searching for a proof of the following fact. Let $A$ and $B$ be C$^\ast$-algebras such that $A$ is a subalgebra of $B$ (in the C$^\ast$-algebraic sense of course) and Let $C = \overline{...
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1answer
41 views

Doubly infinite matrices $A=(a_{i,j})_{i,j=\infty}^{\infty}$

Let $A=(a_{i,j})_{i,j=\infty}^{\infty}$, where $$ \|A\|:=\sum_{r=-\infty}^{\infty}\sup_{j}|a_{j,j+r}|<\infty. $$ I want to show that for all matrices $\|AB\|\leq\|A\|\|B\|$. I obverse that $$ (AB)...
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0answers
18 views

noncommutative algebra

If we let $B$ be a noncommutative Banach algebra with unit $e$, then, obviously, $xy\not=yx$, but are they related? I suspect that the spectral radii of $xy$ and $yx$ are the same, but I couldn't ...
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1answer
12 views

If $f \in L^p(\epsilon, T)$ for every $\epsilon > 0$, is $f \in L^p(0,T)$?

If $f \in L^p(\epsilon, T)$ for every $\epsilon > 0$, is it necessarily true that $f \in L^p(0,T)$? I don't see why not since the only point we have a problem may be at 0, but that is a null set. ...
3
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1answer
31 views

Algebra $A$ and its Gelfand spectrum

Let $A$ be the set of all function $f$ on $\mathbb{R}$ of the form $$ f(x)=d+\int\limits_{0}^{\infty}e^{ixt}k(t)dt,\qquad\quad x\in\mathbb{R}, $$ where $d\in\mathbb{C}$ and $k\in L_1([0,\infty])$. The ...