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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Best approximation. Hahn Banach theorem.

Let $A$ be a normed space over R. Let $B$ be a proper closed subspace of $A$. If $a_0 \in A$ and $b_0 \in B$, $||a_0-b|| \geq ||a_0 - b_0||$ for all $b \in B$ if and only if there is a $f \in V^*$ ...
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1answer
14 views

How does this inequality of a complex function hold

I cannot figure out how $\Re[g(\lambda)]\leq |\lambda|$ implies $|g(\lambda)|\leq|2 r-g(\lambda)|$ where $\lambda$ is an arbitrary complex number s.t. $|\lambda|\leq r$, and $g$ is an entire function. ...
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1answer
30 views

Is this a linear functional?

$A$ is a normed vector space and $B$ is a closed subspace of A. Let $\phi \in V^*$ For $a \in A$, $ \phi (a) = inf${$|| (a - b)|| :$ for all $b \in B $ } I need this to be true for my argument ...
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2answers
23 views

Using the denseness of trigonometric polynomials to prove the following

$f:[0,2\pi] \to \mathbb R$ is a continuous function. For every trigonometric function $T(x)=\sum_{k=0}^n a_k\cos(kx)+b_k\sin(bx)$, we have $\int_0^{2\pi}f(x)T(x)dx=0$. We need to prove $f=0$(and I ...
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1answer
43 views

When can I use Hahn-Banach theorem.

Given a smooth function $v$ with compact support, we could define a linear functional $f: C_c^1(\mathbb R) \rightarrow \mathbb R$ $$f(u) = \int v' u'$$ and we see that $f$ is continuous with respect ...
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1answer
41 views

A question on the continuity of a functional

Suppose $u \in L^{p}(\Omega)$, $\Omega$ is a bounded subset of $\mathbb{R}^n$. Let $q+1<p$ and $p \geq 2$. Is the functional defined by $v\mapsto\int_{\Omega}u^qv$ continuous over ...
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0answers
53 views

a question of my real analysis class,can someone help me solve this question?

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
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1answer
24 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 ...
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2answers
15 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) ...
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2answers
32 views

To show that L^infinite space does not have a countable dense set. [duplicate]

I was able to show that when p>=1, the Lp space on the interval [0,1] has a countable dense set. However, when p is infinite, how to prove that Lp space on the interval [0,1] does not have a countable ...
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1answer
37 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
30 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
32 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
36 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 ...
0
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1answer
24 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 ...
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0answers
20 views

Upper bound for the ratio of Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex and $z$ is a positive real number. Do you know any results about it? Thank ...
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0answers
40 views

Bounds for spectrum of self-adjoint operator on Hilbert space

$A$ is an self-adjoint bounded operator on Hilbert Space $H$, that is for all $x,y\in H$, $(Ax,y)=(x,Ay)$. $(~,~)$ is inner product of H. $$ m=\inf\limits_{||x||=1}(Ax,x) ~~~~~ ...
3
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1answer
23 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
53 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 $A$ ...
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0answers
23 views

Approximate eigenvectors of the closure

Let $A_0$ be a closable operator on a Hilbert space and let $A = \bar A_0$ (i.e. $A = A_0^{\ast \ast}$). Let further $(f_n) \subset D(A)$ (domain of $A$) be an approximate eigenvector for $z \in ...
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2answers
35 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 $f_b((a_i)_{i=0}^\infty) = \sum_{i=0}^\infty a_ib_i$. ...
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1answer
31 views

Tangent vectors to a space of parametrization maps and vector fields

In studying problems of locomotion of deformable bodies in highly viscous fluids I found something that although I have an intuition about I don't know how to make it mathematically rigorous. The ...
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0answers
20 views

Banach-Alaoglu theorem for dual pairs

The versions of the Banach-Alaoglu theorem that I know of always concern some space $X$ and its topological dual $X^*$. Can the theorem be restated for arbitrary (nondegenerate) dual pairings $(X,Y)$ ...
3
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1answer
28 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 = ...
5
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0answers
31 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
40 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), ...
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0answers
37 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 $$-\Delta u = 0 \text{ on $\Omega$}$$ ...
4
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1answer
65 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
61 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 ...
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0answers
15 views

States: Density

Problem Given a Hilbert space $\mathcal{H}$. Regard the CAR-algebra: $$\{a(\eta),a(\zeta)\}=0\quad\{a(\eta),a(\zeta)^*\}=\langle\eta,\zeta\rangle$$ Consider a density: ...
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1answer
77 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 ...
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2answers
34 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 $x_{0}\in O_{1}$,$rank(f)(x_{0})$ ...
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1answer
47 views

Linear and nonlinear operator on normed space and its properties

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 equivalent in linear ...
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0answers
23 views

Non-vanishing integral connected with harmonic functions

Reed and Simon in their book "Functional Analysis" in Section VI.5 show an application of the theory of compact operators to existence of harmonic functions. Their argument is somewhat sketchy and ...
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0answers
50 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 ...
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0answers
15 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 ...
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1answer
37 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
32 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 ...
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4answers
100 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
15 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 ...
4
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0answers
45 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 ...
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0answers
16 views

Example of an ideal in $C(\Bbb D)$ that is not self adjoint

Give an example of an ideal in the C*-algebra $C(\Bbb D)$ that is not self adjoint. My attempt: The function $f: \Bbb D \to C$ such that $f(t) := t+i$ belongs to $C(\Bbb D)$. Let I be the ideal ...
3
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2answers
99 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 ...
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2answers
31 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
67 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 ...
5
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0answers
43 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
523 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 ...
1
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0answers
17 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
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1answer
44 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 ...
2
votes
2answers
69 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 ...