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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33 views

Domain of $\left(f(x)\right)^a$ where $a$ is an irrational number.

Why, if $f(x)$ is a real function and $$\left(f(x)\right)^a$$ where $a$ is an irrational number, we put $$f(x)>0$$ for its domain?
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28 views

Why does one only consider one-parameter groups in Borchers-Arveson theorem?

The theorem (Operator algebras and Quantum statistical mechanics vol. 1, Bratteli, Robinson, Thm. 3.2.46 p.261) roughly says that if one has a one parameter automorphism group $t \rightarrow\alpha_t$ ...
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1answer
14 views

Is linear space a banach space?

Is $(X, || . ||)$ with $X = C([0,1])$ and $||f|| = \left( \int_0^1 |f(s)|^{3} dx \right)^{\tfrac{1}{3}}$ a banach space? It is obvious that this norm is homogeneous and $X$ is a linear space, but I ...
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24 views

Approximating $u \in H^1$ s.t. $u(T)=0$ with $u_n \in H^1_0$ in the gradient norm?

Let $u \in H^1(0,T)$ with $u(T)=0$. Is it possible to find a sequence $u_n \in H^1_0(0,T)$ such that $\nabla u_n \to \nabla u$ in $L^2$? I only need the convergence in the gradient.. not the full ...
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36 views

Showing that if any non-zero $f \in X^*$ takes its maximum value on the unit sphere at most once, then X is strictly convex

Let $(X, \| \|)$ be a normed space. I'm trying to show that if all non-zero $f \in X^*$ take their maximum value on the unit sphere at most once, i.e. $\forall f \in X^* - \{0\}$ there is at most one ...
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11 views

Uniqueness of Best Approximation. Chebyshev Theorem.

Let $X = [0,1]$ Let $V = C(X)$, the space of continuous real-valued functions equipped with supremum norm. Let $P_0$ be the subspace of $V$ consisting of constants. Prove that for $f \in V$, its ...
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32 views

Uniqueness of best approximation. (the sketch of the proof)

Let $X$ be a compact Hausdorff space. Let $A = C(X)$, the space of real-valued continuous functions with supremum norm. Prove : if $X$ has at least 2 points, then there is a one-dimensional subspace ...
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31 views

Møller Operators: Functional Calculus

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider Hamiltonians: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ $$K:\mathcal{D}(K)\to\mathcal{K}:\quad K=K^*$$ and a bounded ...
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147 views

An interesting semi-linear PDE problem

Assume $u\in H^1(\mathbb{R}^n)$ has compact support, and assume that it is a weak solution of the semi linear equation $$ -\Delta u+c(u)=f\;\;\text{in}\;\mathbb{R^n} $$ where $f\in ...
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1answer
23 views

Prove the set is closed with respect to its norm…

Let $V$ be a normed vector space over R. Let $W$ be a proper closed subspace of $V$. We say $w^*$ is a best approximation in $W$ to $v^* \in V$ if $\|v^*-w^*\| \leq \|v^*-w\|$ for all $w \in W$. ...
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24 views

Heat semigroup inequality

Let $M$ be a complete Riemannian manifold and $f \in L^p(M)$. I was wondering if there is any relation between $e^{t\Delta} u^p$ and $(e^{t\Delta}u)^p$, that is, if one is always less or greater than ...
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34 views

When is the Laplace Beltrami Operator self-adjoint?

The Laplace-Beltrami operator is an operator which is the typical example of a self-adjoint operator in $L^{2}$. I am wondering if this is also true for other Hilbert spaces $W^{k,2}$. If this is ...
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29 views

$d(x,L)=\max\{f(x) \,| \, f\in L^{\perp},\, \|f\|=1\}$

Let $X$ be a normed space and $L$ its subspace. Let $L^{\perp}$ be a set of all functional of whose kernel contains $L$. Then $d(x_0,L)=\max\{f(x_0) \,| \, f\in L^{\perp},\, \|f\|=1\}$ I read a ...
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1answer
53 views

If $f \in L^1(M)$, is it true that $f(x) < \infty$ for almost all $x$?

If $M$ is a measurable space (eg. $M$ is a Riemannian manifold which is compact) and if $f \in L^1(M)$, is it true that $|f(x)| < \infty$ for almost all $x$? I am trying to figgur out if $u \in ...
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17 views

the set of Best Approximation is a norm-closed convex set.

Let $V$ be a normed vector space. Let $W$ be a proper closed subspace of $V$. Let $M$ be the set of best approximations in $W$. Prove that $M$ is a norm-closed convex set. I've shown that the ...
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49 views

All infinite dimensional vector spaces with a countable basis are space of sequences or functions?

I'm searching a proof that every infinite dimensional vector space that has a countable basis (Schauder basis), can be represented as a space of functions or sequences. All vector spaces of this ...
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1answer
32 views

A class of functions dense in $L^2$

Suppose $f\in L^2([0,1],\Sigma,\mu)$. Is the class of all $$f=\sum_{i=1}^n \alpha_i (\chi_{A_i}-\chi_{[0,1]/A_i} )$$$A_i\in \Sigma$ to be dense in $L^2([0,1],\Sigma,\mu)$? Thanks.
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31 views

Determine whether the set is closed or not

Suppose $B = \{(x_n)_x \in l^2 \mid \sum_{n=1}^{\infty} |x_n| \leq 1\}$. Check if $B$ is closed in $l^2$ topology (by $l^2$ I mean sequences such that $\sum_{n=1}^{\infty} x_n^2 < \infty$). I'm ...
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1answer
18 views

How to prove a set is norm-closed?

I have to prove that the given space is 'norm-closed convex.' I proved the 'convex' part. But I don't know how to prove a set is 'norm-closed' I think I have to do the followings. Let X be a ...
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18 views

Finding the adjoint of differential operator [closed]

You can find the information about my request is this picture
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1answer
22 views

Relation between Lipschitz condition and linear growth condition

If for a function $f:\mathbb{R}\rightarrow\mathbb{R}$ it is given that it satisfies a Lipschitz condition $\big|f(x)-f(y)\big| \le L\big|x-y\big|$, for all $x,y\in\mathbb{R}$, can we say anything ...
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2answers
37 views

A dense subspace of L^2

Let $\mathcal{H}$ be the Hilbert space of holomorphic functions defined on the unit disc $D\subset\mathbb{C}$ which is the clousure of the complex polynomial functions on the disc with respect to the ...
2
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1answer
23 views

On closed ranges and sequences which converge to zero

I'm reading a proof of the Fredholm alternative, and there is a claim that goes like this: Let $K:X\rightarrow X$ be a compact linear map. Define $T=I-K$, then $Y=\ker(T)$ is a finite dimensional ...
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11 views

Strictly positive element in a C*-algebra

Searching about strictly positive elements, I found this exercise. I tried to solve it, and the following is my attempt. Please check my proof. Is it correct? Suppose $a$ is strictly positive. By ...
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1answer
33 views

How do I show that the sequence below is a Cauchy sequence?

If $(N, \|.\|)$ is a normed space and $(x_n)$ a sequence in $N$ such that $\|x_n - x_{n+1}\| < \frac{1}{2^n}$ Then $(x_n)$ is a Cauchy sequence. Just a hint on how to prove this ...
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0answers
25 views

Variational optimization problem with several constraints

I am looking for solutions, approaches or hints to solve this variational optimization problem: Let $f:\mathbb{R}\rightarrow [0,\infty)$ be such that $\int f(x)\,dx=1$ and $\int x\,f(x)\,dx=0$ and ...
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0answers
33 views

A measure on the space of probability measures

I've been reading about optimal transport and it's connections to geometry. At some point one has to study a bit of the structure of the space of probability measures, $\mathcal{P}(X)$, (over a metric ...
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0answers
10 views

amenable groups versus amenable graphs

In operator algebras, one is often concerned with amenable groups, defined by one of many equivalent conditions. http://en.wikipedia.org/wiki/Amenable_group#Equivalent_conditions_for_amenability In ...
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1answer
30 views

Corollary of Hahn Banach Theorem (quite simple question?)

The corollary of Hahn Banach theorem states : If $E$ is a normed space and $x \in E$ is a nonzero element, then there exists $\alpha \in E^*$ with $|| \alpha || = 1$ and $\alpha (x) = ||x||$ Proof ...
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1answer
28 views

Duality (conjugate) function for $f \in L^\infty$

For $f\in L^\infty(\mathbb{R})$, can I find the $f^*\in (L^\infty(\mathbb{R}))^*$ such that $$\|f^*\|_* = 1 \text{ and } \langle f^*, f \rangle = \|f\|_\infty.$$ I know that ...
0
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1answer
32 views

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
29 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 ...
3
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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
52 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
35 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
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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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
52 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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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 ...