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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28 views

A question about the natural identification between $Y$ and $Y^{**}$, with $Y$ normed space. Is the following fact obvious?

Let $X$ be a reflexive normed space, $Y$ a normed space and $T: X \to Y$ a linear operator. I consider a sequence $\{x_n\}$ in X and the sequence $\{T(x_n)\}$ in $Y$. I know that there is a ...
0
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2answers
16 views

behavior of function between two bounds

Let $f, U, L : [0,1] \rightarrow \mathbb{R}$ be three functions with the property that (1) U and L are continuous functions (2) $\forall x \in [0,1]$, $L(x) \leq f(x) \leq U(x)$ (3) $f(0)=L(0)=U(0)=...
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2answers
105 views

Show that $L^{\infty}$ space does not have a countable dense set. [duplicate]

I was able to show that when $p ≥ 1$, the $L^p$ space on the interval $[0,1]$ has a countable dense set. However, when $p$ is infinite, how to prove that $L^p$ space on the interval $[0,1]$ does not ...
1
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1answer
45 views

A question about $n$-dimensional operator space

Let $F_{n-1}$ be the free group of rank $n-1$ and $C^{*}(F_{n-1})$ be the universal group C*-algebra of $F_{n-1}$. And if $E_{n}$ is the $n$-dimensional operator space in $C^{*}(F_{n-1})$ spanned by ...
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1answer
38 views

Cardinality of the set of linear discontinuous functionals in a normed space

How does one show (or disprove) that for any infinite-dimensional normed vector space $V$, there are uncountably many linearly independent elements in $V^{*}\setminus V'$, where $V^{*}$ and $V'$ ...
4
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0answers
41 views

Prove that $\dfrac{g(x,u_{n})}{\left\Vert u_{n}\right\Vert ^{p-1}}\rightarrow g_{0}$ weakly in $L^{\overline{p}}$ for some $\overline{p}>p*'$

Let $\Omega \subset \mathbb{R}^{N}$ be a smooth bounded domain , $g:\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ is a Caratheodory function such that $g(x,t)=0$ for $t\leq0$ . Suppose that ...
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0answers
145 views

Hilbert-Schmidt and compact operators

I am new to this site and i dont really know how to ask questions properly, so i am really sorry if i did something wrong. My question is if there is a way to prove that a Hilbert-Schmidt operator is ...
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1answer
41 views

Modular group maps upper half to itself in complex plane

Let $U$ is upper half complex plane: Suppose $$H=\{{{az+b\over cz+d}:a,b,c,d \in \Bbb R, ad-bc \gt0}\} $$ be set of modular group. Now I have to prove $H=Aut(U)$ I have some ideas, I was trying to ...
3
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1answer
26 views

A trouble with the existence of an $C_{0}^{\infty}$- function $v,v>0$ such that $\int_{\Omega}hv^{\alpha}>0$

Let $\Omega$ be a smooth bounded subset of $\mathbb{R}^{n}$ , an $L^{\sigma_{\alpha}}$ -function $h$ with $h^{+}\neq0$ , $\dfrac{1}{\sigma_{\alpha}}+\dfrac{\alpha}{p*}=1$ , does there exist ...
1
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1answer
63 views

Cauchy sequences on certain set

Suppose $(X,d)$ be a metric space. Let $(a_n)$ be a sequence in $X$ such that $(a_{n})$ has no Cauchy subsequence. Let $A=\{a_{n}:n\in\mathbb{N}\}$, is it true every Cauchy sequence $(b_n)$ in $A$ ...
0
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2answers
44 views

Show a map from $(c_0)^*$ to $\ell^1$ is an isometry

So, for a bit of context, i'm trying to show that $(c_0)^* = \ell^1$. Given $b = (b_i)_{i=0}^\infty \in \ell^1$, define $f$ in $(c_0)^*$ by $f_b((a_i)_{i=0}^\infty) = \sum_{i=0}^\infty a_ib_i$. ...
3
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1answer
38 views

Identity Operator can be uniformly approximated by orthonormal basis

Let $H$ be a separable Hilbert space with orthonormal basis $e_1, e_2, ...$. I know that for any $x \in H$, we have $$\|x\|^2 = \sum\limits_n \|\langle x, e_n \rangle\|^2$$ and in fact $x = \lim\...
5
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1answer
55 views

Spectrum of periodic schrödinger operators

In many articles it's stated, as if it's common knowledge, that any Schrödinger operator with periodic potenial has purely absolutely continuous spectrum. I've tried to actually find a theorem ...
1
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1answer
65 views

A paradox derived from the open mapping theorem

The problem comes from Erwin Kreyszig's Introductory Functional Analysis with Applications, section 7.4, problem 4: Let $T:l^2\mapsto l^2$ be defined by $y=Tx, x=(\xi_j), y=(\eta_j), \eta_j=\...
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0answers
56 views

Existence of weak solution to $-\Delta u =0$, $u|_{\Gamma_1} = g$ and $u_{\Gamma_2} = 0$?

Let $\Omega$ be a compact Riemannian manifold with $\partial\Omega = \Gamma_1 \cup \Gamma_2$ a union of disjoint sets. Let $g \in H^s(\Gamma_1)$ and consider $$-\Delta u = 0 \text{ on $\Omega$}$$ $$u|...
4
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1answer
79 views

$\int_a^b |f(x)||g(x)| dx \leq \left(\int_a^b |f(x)|^p dx\right)^{\frac1p}\left(\int_a^b |g(x)|^q dx\right)^{\frac{1}{q}}$

Let $p\gt 1,q\gt 1$ be the dual indices, $\frac1p + \frac1q = 1$ and let $X$ be the space of all continuous functions on $[a,b]$ with two real numbers $a\lt b$. $f(x)$ and $g(x)$ are continuous ...
2
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1answer
67 views

Solutions to $u \circ v - v \circ u = \mathrm{Id}$

Let $(E,\Vert \cdot \Vert)$ be an infinite-dimensional normed vector space and $\mathcal{L}_{c}(E)$ denote the ring of continuous endomorphisms of $E$. I would like to determine whether the equation ...
0
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1answer
132 views

Help with an inequality in Cazenave's book “Semilinear Schrodinger equations”

I'm reading Cazenave's book "Semilinear Schrodinger equations" and I found this inequality at page 84 $$\vert\vert u_1\vert^\alpha u_1-\vert u_2\vert^\alpha u_2\vert\vert\leq C (\vert u_1\vert^\alpha+\...
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2answers
49 views

a question about functional analysis conclusion,and I am not sure whether it is true or not?

we have $R^n$,$R^m$ spaces, suppose open set $O_{1}\subset R^n $ and $O_{2}\subset R^m$, $f:O_{1}->O_{2} $ is k-times differentiable$(1<=k<=\infty)$,then at $x_{0}\in O_{1}$,$rank(f)(x_{0})$ ...
1
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1answer
125 views

Linear and nonlinear operator on normed space and its properties

My first question is : We know every bounded linear operator is continuous operator , and every continuous linear operator is bounded operator , in other words the continuousness and boundedness are ...
4
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0answers
75 views

Proving completeness of $L^p$

I want to make sure my understanding of the proof is correct. For a Cauchy sequence $\{f_n\}$ in $L^p$, we want to find a $f\in L^p$ such that $f_n\stackrel{L^p}\to f$ Now, skipping the ...
1
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0answers
32 views

Calculus of Variations: What if the functional is an integral with boundaries at infinity?

I am trying to grasp the basics of Calculus of Variations. The problem seems to be concentrated on functionals of the form : $$ F[y] = \int_{a}^{b} G(y,y(x),y'(x))dx$$ where $y(x)$ is assumed to be ...
2
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1answer
63 views

Trace operator on $W^{1,\infty}$

My question is rather short and simple really: Is the trace operator well defined on $W^{1,\infty}(\Omega)$ for some bounded Lipschitz domain $\Omega$? The reason I ask is because I have seen ...
0
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1answer
43 views

weak convergence and continuity

Suppose $F:X \rightarrow X'$, $X'$ being the dual space of the normed linear space $X$, is a continuous map. Let $\{x_n\}$ be a sequence in $X$ which converges weakly to $x$ in $X$. Can I conclude ...
7
votes
4answers
210 views

Use of $L^2$ norm in calculus of variations

I am trying to make an introduction to the calculus of variations. This field has many connections with functional analysis, in which I do not have an experience. I recently learned about function ...
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0answers
24 views

Subclasses of simple functions dense in $L^2$

Q1. Consider $f\in L^2([0,1], R)$ with $ ||f||^2=\int f(x)^2d\mu(x)$ Consdier a subclass of simple functions $f= \sum_{i=1}^n a_i \chi_{A_i}$ where $A_i\in \Sigma$ (on $[0,1]$) and $A_n\subseteq...\...
4
votes
1answer
111 views

Modifying a smooth function with respect to conditions on its partial derivates

Let $\{U_i\}_{i\in I}$ be a locally finite collection of open subsets of $\mathbb{R}^n$, $K_i\subseteq U_i$ compact subsets, $\epsilon_i>0$ positive real numbers and a nonnegative natural number $k$...
3
votes
2answers
118 views

Weakly convergent in different spaces

Given $\Omega \subset \mathbb{R}^n$ open connected and $k\geq 0$. Let $f_n, f$ be distributions such that $$f_n \rightharpoonup f \in \mathcal{D}'(\Omega)$$ as $n\to \infty$. Assume that $f_n \in H^k(...
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2answers
45 views

Distance preserving function on a Hilbert space

Let $\Bbb F = \Bbb R$. Show that every preserving function $f$ on Hilbert space $H$ has the form $f(x) = f(0) + Tx$ for some isometry $T$ in $B(H)$. If $f$ is linear then $f$ is an isometry. Suppose $...
4
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0answers
110 views

Are “most” operators on an infinite-dimensional complex Banach space “diagonalizable”? [closed]

This is true for finite-dimensional spaces, of course. To be precise, let $T$ be an operator on a complex Banach space $X$ which is not finite-dimensional. For each $\lambda \in \mathbb{C}$, let $V_\...
5
votes
1answer
57 views

Isomorphism on dense subset

I am wondering if the following could be done. I want to show two Banach spaces $X$ and $Y$ are isomorphic. If $A$ is dense in $X$, and $B$ is dense in $Y$, is it sufficient to show there is an ...
29
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2answers
634 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
0
votes
0answers
21 views

If $||T_n(x)|| \to 0$ pointwise, then $||T_n \circ S|| \to 0$ when $S$ is a compact operator

Here $T_n, S$ are bounded linear operators on a Hilbert space $H$. I have already proved the assertion when $S$ is an operator of finite rank. And when $S$ is compact, one can find a finite rank ...
1
vote
1answer
103 views

how to show the derivative of the polynomial is bounded by itself in certain space.

How to prove that for every positive integer $d$, there exists $C(d)>0$, such that: For every polynomial with degree $\leq d$, we have $\max\limits_{x\in [0,1]}|p'(x)|\leq C(d)\max\limits_{x\in [0,...
2
votes
2answers
89 views

Usefulness of Functional analysis

I heard that functional analysis can be applied to many problems in signal processing. I'm trying to explain to my engineer friend why it is useful, but I learnt it in a pure math setting. Can anyone ...
2
votes
0answers
97 views

Uniform convergence and equicontinuity

Given a sequence of functions which is not uniformly convergent, can we deduce, that none of its subsequences is uniformly continous and therefore, by Arzela-Ascoli say that the family of function is ...
0
votes
3answers
173 views

If the image of a linear transformation of normed spaces is finite dimensional, is the map bounded?

Let $V, W$ be normed spaces. If $T: V \rightarrow W$ is such that $T(V)$ is finite dimensional, does it follow that $T$ is bounded? Edit: This isn't a homework question, I'm just asking because I'm ...
1
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1answer
90 views

Separable spaces isomorphic

I am reading a proof and it states the following without proof: Two separable Banach spaces $X$ and $Y$ are isomorphic iff there are sequences $(x_n)\subset X$ and $(y_n)\subset Y$ such that $\...
2
votes
1answer
86 views

About Cauchy sequence

Suppose $(X,d)$ be a metric space. Let $(a_n)$ be a sequence in $X$ such that $(a_{2n-1})$ and $(a_{2n})$ has no Cauchy subsequence. Is it also true that $(a_n)$ has no Cauchy subsequence? Let $A=\{...
2
votes
1answer
47 views

Why is every $H^1$ function on the circle this way?

I want to know, specifically, why is every $H^1$ function which is defined on the circle a absolutely continuous function, with square-integrable derivative defined almost everywhere. I have no ...
1
vote
1answer
92 views

Dominated convergence theorem and fundamental lemma

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= \int_{\...
1
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1answer
40 views

Dominated convergence and fundamental lemma of the calculus of variation

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= \int_{\...
0
votes
1answer
121 views

Spectrum of simple multiplication operator on $L^2(0,1)$

I'm trying to calculate the spectrum of the linear operator $T: L^2(0,1) \to L^2(0,1)$ given by $T(f) \to tf(t)$. I've found a few facts about this operator but I'm still struggling to find the exact ...
3
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0answers
147 views

Complicated convergence of nonlinear term

Let $1<p<\infty$, $\Omega\subset\mathbb{R}^m$ be open, bounded with $\partial\Omega\in C^1$. Assume that $u_k\to u$ weakly in $W^{1,p}(\Omega;\mathbb{R}^n)$. We know that $u_k\to u$ strongly in $...
1
vote
1answer
136 views

show Lebesgue dominated convergence theorem fails for ${n^2xe^{-nx}} x\in [0,1]$

show Lebesgue dominated convergence theorem fails for the sequence of functions $f_n=n^2xe^{-nx}$ $x\in [0,1]$ Here is my solution. Is it correct? $f_n$ is an integrable function the sequence ...
6
votes
1answer
173 views

How to prove this multivariable function is constant?

Suppose the multivariable function $z=f(x,y)$ is defined on $\mathbb R^2$, has continuous partial derivatives and always satisfies $$x\frac{\partial f}{\partial x}(x,y)+y\frac{\partial f}{\partial y}(...
0
votes
1answer
77 views

Prob. 3, Sec. 2.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $C[-1,1]$ denote the normed space of all (real or complex-valued) functions defined and continuous on the closed interval $[-1,1]$ on the real line, with the norm given by $$\Vert x \Vert_{C[-1,1]}...
0
votes
1answer
32 views

Can we expect $\|fg\|_{\mathcal{F}L^{1}} \leq C \|f\|_{L^{2}(\mathbb R)} \|g\|_{\mathcal{F}L^{1}}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
-2
votes
1answer
84 views

How to prove the function $f(T)=\sum_0^\infty a_n T^n$ is infinitely differentiable [closed]

Let $X$ be a Banach space and let $\mathcal{L}(X)$ denote the Banach space of continuous linear endomorphisms $X\to X$. If the radius of convergence of $\sum_0^\infty a_n z^n$ is $\ge R$, then prove ...
0
votes
1answer
103 views

Show that the norm $\|\cdot\|_{1}: l_1 \rightarrow \mathbb R$ is not differentiable at any point of $l_1$.

Let A be the set of all sequences {${{x_{n}}}$} of real numbers such that $\sum_{n=0}^{\infty} |x_{n}|<\infty$. Let $$\| {{x_{n}}} \|_{1}=\sum_{n=0}^{\infty} |x_{n}|.$$ Show that the norm $\|\...