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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170 views

Show that $\log(x)$ is a Bounded Mean Oscillation (BMO)

As an extension of our class notes, we were asked to show that the function $w =\log(x)$ is a Bounded Mean Oscillation (BMO). First off, I believe our professor made a mistake, and really wanted us ...
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1answer
26 views

Bases in Hilbert space

There's a theorem that states that having Hilbert space $H$, orthonormal basis $\{x_n\}$, and a set of linearly independent unit vectors $\{y_n\}$, such that $\sum\limits_{n=1}^{\infty}\|x_n - ...
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47 views

Resolvent of the differentiation operator on the torus

Suppose we have the space $L^2(\mathbb T)$, that is, the space of periodic functions that are in $L^2$. Let $L^0$ be the operator of differentiation, i.e $L^0 f = f'$ where the domain of $L^0$ is the ...
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1answer
48 views

Proof: In a topological vector space, every neighborhood of $0$ contains a balanced neighborhood of $0$

I was reading this proof in Rudin 2/e (Th 1.14), but couldn't work it out. Suppose $U$ is a neighbhorhood of $0$ in the topological vector space $X$, then Since scalar multiplication is ...
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1answer
45 views

Examples of functions with values in distributions

What is an example of a function in $L^p((0,T);\mathcal{D}'(\mathcal{R}))$? I ask this because the Majda-Bertozzi book on Incompressible flow deals with vortex sheet initial data $\omega(t,\cdot)\in ...
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1answer
56 views

Convergence of function in $L^1$ space

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
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2answers
68 views

Sequence of unit vectors in a Hilbert space

Question: Let $H$ be a Hilbert space and $\{\xi_{i}\}\subset H$ be a sequence of unit vectors. Suppose that $||T_{j}(\xi_{i})-\xi_{i}||\rightarrow0$ as $i\rightarrow\infty$, for $j=1, 2, ...n$ (here ...
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31 views

Question about Green function

how to find the Green function of this problem : $$ \begin{cases} -(p(t)u'(t))'=\lambda f(t,u(t)) ~ \text{a.e.} ~t>0\\ u(0)=u(+\infty)=0 \end{cases} $$ Thank you
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53 views

bilinear form, anti symmetric part

$\mathcal{H}$ : real Hilbert space with inner product $(\,,\,)$ and norm $||\,||:=(\,,\,)^{1/2}$ Let $D$ be a linear subspace of $\,\mathcal{H}$ and $\mathcal{E}$ : $D\times D\to \mathbb{R}$ a ...
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1answer
82 views

Are the following norms equivalent?

We have the norms $||f||_1=||f||_\infty+||f'||_\infty$ and $||f||_2=|f(a)|+||f'||_\infty$ where $f\in C^1[a,b]$. Are they equivalent and how shoud I prove/disprove this.
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45 views

Question about injection on an unbounded space

I have this space $$C_0((0,+\infty))=\left\lbrace u,u\in C((0,+\infty)),\lim_{t\rightarrow +\infty} u(t)=0\right\rbrace$$ with the norm $$||u||_{\infty}=\sup_{t\geq0}|u(t)|$$ how to prove that ...
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141 views

These germs make me sick!

I need a "mini-crashcours" concerning the space of germs of continuous functions in order to solve an exercise which requires me to show that limits in this space aren't always unique. We have ...
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164 views

Example of weak derivative on multivariable function

In order to explain about the concept of weak derivatives, I plan to give examples on them. I already manage one example for the single-variable case, but I think it would be better if I can provide ...
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3answers
116 views

Showing that a sequence converges in norm

I am trying to prove the following: Let $X$ be a normed linear space satisfying the property: $\forall \left\{x_n\right\}, \left\{y_n\right\} \subseteq X $, we have $\|x_n\|=\|y_n\|=1, ...
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2answers
79 views

showing uniqueness of a Hahn Banach extension

I am trying to prove the following: If $H$ is a Hilbert space and $G\subseteq H$ is a closed linear subspace, then any bounded linear functional on $G$ has a unique Hahn-Banach extension on $H$. So ...
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1answer
64 views

An exercise about projections on Hilbert space

Let $H$ be a Hilbert space with an orthonormal basis $\{v_{n}\}_{n=1}^{\infty}$. The C$^{*}$-algebra $K$, the set of all compact operators on $H$, is a non-unital C$^{*}$-algebra. Let $p_{n}$ be the ...
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1answer
37 views

Is the particle in a ring a regular Sturm-Liouville problem?

The problem of a particle in a ring is a well-known eigenvalue problem $$\frac{d^2}{d\theta^2} \psi(\theta) + V_0 \psi(\theta) = \lambda \psi(\theta)$$in physics and the Schrödinger equation has a ...
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1answer
62 views

Equivalence of norms given continuous identity

It is known that $\parallel \; \parallel_{1}$ & $\parallel \; \parallel_{2}$ are equivalent norms over $X$ if there are $A,B>0$ such that $A\parallel x \parallel_{1} \leq \parallel x ...
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98 views

Show that $T$ is continuous

I have a question about how to response to: Given a Banach space X and $T: X \rightarrow X^{*}$ a linear operator such that $\langle Tx,x\rangle \geq 0$ for all $x \in X$. Show that $T$ is ...
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1answer
53 views

Bound on Bessel potential

Let $s\in\mathbb{C}$. For a complex number $z$, $Re(z)>0$, consider the Bessel potential $$K_s(z)=\int_0^{+\infty}e^{-z\cosh t}\cosh(st)dt$$ I need to prove that, if $|z|\leq 1$, then ...
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1answer
31 views

Approximating a mean on $L^{\infty}(G)$ by a norm $1$ non-negative $f\in L^{1}(G)$

Let $G$ be a locally compact group. A mean $M$ on $L^{\infty}(G)$ is a continuous linear functional on $L^{\infty}(G)$ such that $1 = \langle 1 , M\rangle = \|M\|$. My Exercise: Show that the set ...
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1answer
109 views

Proof of an identity in a Hilbert space

Let $x_1,x_2\in H$ be two unit vectors in a Hilbert space $H$ and let $t_1,t_2 : B(H)\rightarrow\mathbb{C}$ be the linear functionals given by $t_j(a) =\langle ax_j,x_j\rangle$. Define ...
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48 views

Why is $\overline{L^\infty(\Omega)} \subset L^\infty(\Omega) $ where the closure is in norm of $L^1(\Omega)$?

Let $\Omega$ be a domain which may or may not be unbounded (eg. $\Omega = B_1(0)\times (0,\infty)$). Why is $$ \overline{L^\infty(\Omega)} \subset L^\infty(\Omega)$$ where the closure is in norm of ...
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55 views

Problem of convergence of characteristic functions

Let $\Omega \subset \mathbb{R}^{n}$ be bounded and open. Assume that $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is a Caratheodory function which means $a(x,.,.)$ is ...
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1answer
53 views

Are there necessary and sufficient conditions for Krein-Milman type conclusions?

This, the third of three self-answered questions, contains a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The first question is here. The second question is here. ...
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23 views

Conditions for augmenting a collection of sets so that the new sets are small but the hull is large?

This is the second of three self-answered questions which will culminate in a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The first question is here. The third ...
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1answer
58 views

Finding the Hilbert Adjoint in this case

If we let $H$ be a Hilbert space with inner product $\langle.,.\rangle$. And we fix $y, z \in H$. Then let $T:H\rightarrow H$ be the bounded linear operator $Tx = \langle x,y\rangle z$. Then what is ...
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63 views

Does completing a normed space commute with taking quotients?

Let $X$ be a normed vector space and $Y \subset X$ a closed subspace. We consider the quotient $X / Y$ and equip it with the quotient norm. Then we may form the completion $\overline{X / Y}$. We ...
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1answer
114 views

Maximal monotone operator without convex domain?

I'm looking for an example of a (multi-valued) maximal monotone operator $A$ mapping a Banach space $X$ into its dual $X^*$ such that the domain $D(A)=\{x\in X: Ax\neq\emptyset\}$ is not convex. ...
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1answer
134 views

Properties of this ODE

Given the ODE $$ \{e^{\alpha\sin\theta}\Psi'(\theta)\}'+ e^{\alpha\sin\theta}\beta\cos^{2}\theta \Psi(\theta) = \lambda e^{\alpha\sin\theta}\Psi(\theta). $$ where $\theta \in ...
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80 views

When do partial subgradients give a subgradient?

I'm looking for sufficient conditions that guarantee that partial subgradients of a convex, lower-semicontinuous functional $f:X_1\times X_2\rightarrow\overline{\mathbb{R}}$ form a subgradient of $f$. ...
3
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1answer
91 views

UNBEATABLE recurrence relation

Hi I don't know where to start to solve this reccurence relation: $g(1)=2$ $ g(2n)=3g(n)+1$ $ g(2n+1)=3g(n)-2$ of coures I can make it: $ g(1)=2$ $ g(n)=g(2n)/3-1/3$ $ ...
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33 views

Dual of a Sobolev space (ex)

Consider $f(x_1,x_2)=\chi_{B_1(0,0)}(x_1,x_2)$. 1) Is $\nabla f\in (H^1_0(\mathbb{R}^2;\mathbb{R}^2))^*$? 2) $\langle \nabla f , u \rangle= ?$ with $u\in H_0^1(\mathbb{R^2})$? For the first ...
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1answer
49 views

Are there necessary and sufficient conditions so that every element in a partially ordered set is either the least element or in the upset of an atom?

This is the first of three self-answered questions which will culminate in a proof of necessary and sufficient conditions for Krein-Milman type conclusions. The second question is here. The third ...
2
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2answers
89 views

Equivalent norms, identity function

If $||*||_{1}$ and $||*||_{2}$ are norms on $X$ and $I:(X,||*||_{1}) \rightarrow (X,||*||_{2})$ is the identity function and it is continous... are $||*||_{1}$ and $||*||_{2}$ equivalent? It is ...
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1answer
77 views

Multiplication operator has no no eigenvalues

Statement: Let $M_x$ denote the multiplicative operator acting on $L^2([0,1], \, dx)$ by $M_x(f) = xf$. Show that $M_x$ has no eigenvalues Attempt: Let $M_x(f) = xf$ then we should have $M_x(f) = ...
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1answer
58 views

Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
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1answer
114 views

Strictly convex unit balls in $L^p$

I need to show that if $1<p<\infty$, then the unit ball is strictly convex in $L^p$, that is, $||\lambda x+(1-\lambda)y|| < 1$ whenever $||f|| = ||g||=1$ and $\lambda \in (0,1)$. I tried ...
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108 views

k-times differentiable functions on [0,1]

Is $C^k[0,1]$ (the set of all k-times differentiable function, not necessarily continuously) complete with respect to the norm $\|f\|_\infty + \|f'\|_\infty +\cdots+\|f^{(k)}\|_\infty$? I know the ...
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50 views

$\ell^p\subset\ell^q$ if $1<p<q<\infty$

I need a reference states that $\ell^p\subset\ell^q$ if $1\leq p<q<\infty$. I could find the result on wikipedia and some homework sets but I need to cite this in a paper I am writing.
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202 views

Converse of a fixed-point theorem

I'm having some trouble furnishing a proof here. Let $(E, d)$ be a metric space such that any $k$-Lipschitz function has a fixed point for $0 < k < 1$. Does it follow, then, that $E$ is ...
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34 views

Spectrum of the operator of differentiation along streamlines

Suppose $\mathbb T^2$ denotes the two-torus and suppose $\psi^0$ is a steady state smooth stream function on $\mathbb T^2$ and define $\mathbf u^0 = (-\partial_y \psi^0,\partial_x \psi^0 )$ be the ...
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1answer
79 views

Convergence Radius: Non-Analyticity

Why is a function certainly nonanalytic at some point on the radius of convergence? I mean considering a power series around somewhere and if theres a power series expansion at every point on circle ...
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35 views

an $L^p$ implication

Let $1<p<\infty$ and $x,y\in L^p([0,1])$ such that $\|x\|_p = \|y\|_p = 1$. Then the following implication holds: $$\left\|\frac{x+y}{2}\right\|_p=1\Rightarrow x=y\tag{*}$$ This ...
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25 views

Simultaneous extension and complemented subspace

The following is Exercise 3.13.5 of Conway's Functional Analysis: Let $X$ be a compact set and let $Y$ be a closed subset of $X$. A simultaneous extension for $Y$ is a bounded linear map $T:C(Y)\to ...
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1answer
92 views

When is the closed unit ball $B^*$ in the dual space strictly convex?

I'm finding the conditions (on the primal normed space $X$ or on the closed unit ball $B$ of $X$) to ensure that the closed unit ball $B^*$ in the dual space $X^*$ is strictly convex. Anyone can help ...
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1answer
26 views

Why it is bounded?

my question is very simple. It is Given $X$, $Y$ Banach Spaces and a sequence of bounded linear operators $T_{n}$ such that $T_{n} x$ converges to $T x$ in $Y$ for all $x \in X$. I proved that T is ...
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0answers
36 views

Linear map in Hilbert space.

If you have a linear map $h\mapsto T(h)$ from $H_1$ a real separable space, to Hilbert space $H_2$, it seem that this maps provides an isometry of $H_1$ onto a closed subspace of $H_2$. I try to ...
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1answer
54 views

Proposed proof for Sobolev space result

I have the following result which seems that it must be true, but I would like to prove it: This is my proposed proof. If $U \subset \mathbb{R}^{n}$. Given $u \in W^{1,p}(U)$, where $u$ has compact ...
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1answer
46 views

Periodicity and period of a function

The question is : Let $f(x)$ be a real valued function defined for all real numbers x such that for for some fixed real number $a>0$, $f(x+a)=\frac{1}{2} + \sqrt{f(x)-(f(x))^2}$ and $\frac12\le ...