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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26 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? Actually I want to prove the existence of entropy solution for a nonlinear degenerate PDE for an equation that cannot solve ...
1
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1answer
49 views

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, $T_i^2 = T_i.T_i\to 0$...
10
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0answers
75 views

Is there a probability measure on $[0,1]$ with no subsets with measure $\frac{1}{2}$?

I have a decidedly weird question. Does there exist a probability measure $(\mu, \mathcal{F})$ on $[0,1]$ such that 1) $\mu(x) = 0$ for every $x \in [0,1]$ 2) For every $r \in [0,1] \setminus \...
1
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1answer
57 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 $x \rightarrow 0$ ...
0
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0answers
8 views

A matrix decomposition problem for row/column element order

Indeed, I don't know how to classify this problem, but I try to use matrix to describe it. The problem is that there exists a function $f(x, y)$ and its exact form remains unknown. But I have some ...
2
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0answers
19 views

A reflexive subspace of $\ell_p(c_0)$

Fix $1<p<\infty$ and define the Banach space \begin{equation*}\ell_p(c_0)=\left\{(x_n)_{n=1}^\infty\subseteq c_0:\left(\|x_n\|_{c_0}\right)_{n=1}^\infty\in\ell_p\right\}\end{equation*} endowed ...
0
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1answer
25 views

The composition of a dissipative operator and a positive opeartor is dissipative?

Let the real Hilbert space $H^1(\Omega)$ endowed with its usual inner product, denoted by $\langle ., . \rangle$ and let $A : H^1(\Omega) \rightarrow H^1(\Omega)$ be a dissipative ...
26
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10answers
3k views

What are the applications of functional analysis?

I recently had a course on functional analysis. I was thinking of studying the mathematical applications of functional analysis. I came to know it had some applications on calculus of variations. I am ...
5
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1answer
2k views

$C([0, 1])$ is not complete with respect to the norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$

Consider $C([0, 1])$, the linear space of continuous complex-valued functions on the interval $[0, 1]$, with the norm $$\displaystyle\lVert f\rVert_1 = \int_0^1 \lvert f(x)\rvert \,dx.$$ I have to ...
6
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1answer
4k views

About Banach Spaces And Absolute Convergence Of Series

How to prove the following two assertions: If in a normed space $X$, absolute convergence of any series always implies convergence of that series, then $X$ is a Banach space. In a Banach space, ...
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15 views

on $a$ proof that every normed linear space admits a completion.

One of the proofs that any metric space $M$ has a completion is to use the fact that the space $C_\infty (M)$ of all complex-valued continuous and bounded functions on $M$ is complete with respect to ...
11
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4answers
420 views

Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value?

If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ ...
0
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1answer
37 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. ...
7
votes
3answers
103 views

Trace norm of a triangular matrix with only ones above the diagonal

For $n\in\mathbb N^*$, we consider the triangular matrix $$ T_n = \begin{pmatrix} 1 & \cdots & 1 \\ & \ddots & \vdots \\ 0 & & 1 \end{pmatrix} \in M_{n,n}(\mathbb R) \,. $$ ...
41
votes
2answers
16k views

Limit of $L^p$ norm

Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\lim_{p\to\infty}\|f\|_p=\|f\|_\...
0
votes
1answer
26 views

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?
0
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0answers
32 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 ...
-1
votes
1answer
65 views

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 ...
0
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1answer
29 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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0answers
11 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 ...
1
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1answer
24 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(...
2
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2answers
258 views

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 ...
2
votes
1answer
130 views

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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1answer
43 views

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 ...
1
vote
1answer
33 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 ...
15
votes
3answers
4k views

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 ...
1
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1answer
19 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 ...
1
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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? ...
1
vote
1answer
30 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 ...
2
votes
1answer
28 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 ...
0
votes
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
votes
0answers
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 ...
14
votes
1answer
886 views

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 ...
1
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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.
2
votes
1answer
35 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: ...
2
votes
0answers
36 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 ...
0
votes
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 ...
4
votes
0answers
31 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 ...
10
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0answers
267 views
+500

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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0answers
40 views

Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
2
votes
1answer
46 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,...
4
votes
1answer
49 views

The completeness relation from QM in terms of inner products

I remember from QM that the completeness relation says $$ \sum_{n=1}^\infty |e_n\rangle \langle e_n | = I$$ so that $\langle x\mid y\rangle =\sum_{n=1}^\infty \langle x\mid e_n\rangle \langle e_n \...
21
votes
8answers
2k views

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, ...
0
votes
0answers
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 $C$. ...
0
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0answers
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: \...
3
votes
2answers
49 views

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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2answers
2k views

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?
0
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0answers
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 ...
0
votes
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&...
12
votes
1answer
475 views

Heat Kernel Property

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}\...