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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67 views

Let (Y,d) be a complete metric space, and let G be a family of continuous functions from Y to $\mathbb R$…

Let $(X,d)$ be a complete metric space, and $F$ be a family of continuous functions from X to $\mathbb R$. Suppose that for each $x\in X$ there exists $M_x\in$$\mathbb R$ such that $f(x)\le M_x$ for ...
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35 views

What are $C_b^2 (\mathbb R)$ and $C^{2,1} (\mathbb R × \mathbb R^+ )$?

From a note, for a diffusion process with its transition probability $P(, t|x, s)$, Theorem 1. (Kolmogorov) Let $f (x) ∈ C_b (\mathbb R)$ and assume that $$ u(x, s) := ∫ f (y)P (dy, t|x, s) ∈ ...
2
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2answers
176 views

Does Closed Graph imply Closed Range

Suppose I have a bounded linear operator from a space $X$ to $Y$, both Banach. I know that if D(T) and Ran(T) are closed, then the graph G(T) is closed in $X\times Y$. However is the converse true ? ...
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1answer
72 views

Show $A$ is self-adjoint and $ f= Au$ in weak sense.

Hoi, consider $L_2(\Omega)$ with $\Omega = (0,1)\times (0,1)$ and let $u\in L_2(\Omega)$ be defined as $u(x,y) = 1$ for $x>y$ and $u(x,y) = 0$ for $x\leq y$. Let $A = \partial_x^2 - \partial^2_y$ ...
2
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2answers
318 views

$C^\infty(R^n)$ is a Banach Space when equipped with topology of uniform convergence

Prove $C^\infty(\Bbb R^n)$ is a Banach Space when equipped with topology of uniform convergence. $C^\infty(\Bbb R^n)$ is space of all continuous functions that converge to $0$ at $\infty$. And, the ...
4
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1answer
331 views

Reproducing Kernel Hilbert Spaces for Dummies

I am in the middle of some machine learning paper that states that for function $f$, imposing the norm constraint, $\|f \|=1$, corresponds to an orthogonal projection onto the direction selected in ...
1
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2answers
71 views

Operator norm converging to 0 for certain condition

Let $X$ be a finite-dimensional normed space and $T_n : X \to X$ a sequence of linear operators such that $\lim_nT_nx = 0$ for all $x$ in $X$. Prove that $\lim_n\|T_n\|=0$.
2
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1answer
65 views

Families of vectors in finite dimensional Hilbert space

This is an exercise left for the reader in proof of Proposition 1.12 in Pisier's book "Introduction to Operator Space Theory". Assume that we have $n$ vectors $x_1, \dots, x_n$ in $\ell_2^N$ (where ...
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1answer
52 views

If $v = v_1 + v_2$ and $\lVert v\rVert \leq 1$ then $\lVert v_1\rVert \leq 1$. Why is it true? (Hilbert spaces and orthogonality)

We have $H^1_0 \subset L^2$ where $w_j$ is an orthogonal basis on $H^1_0$ and orthonormal basis on $L^2$. Let $v= v_1 + v_2$ with $\lVert{v}\rVert_{H^1_0} \leq 1$, where $v_1 \in \text{span}\{w_j\}$ ...
3
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344 views

Application of Banach fixed-point theorem

I am looking for a function $f:[0,1]\rightarrow\mathbb R$ which satifies $\int_{0}^{1}\frac{\sin(f(t)-y)}{2}\,\mathrm dy=f(t)$ for $t\in[0,1]$. The first thing I do is to define a function ...
3
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0answers
167 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
3
votes
2answers
198 views

Two proofs concerning Hölder's inequality

I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ...
2
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1answer
2k views

The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. [duplicate]

If we are in a sequence space, then the $ l^{p} $-norm of the sequence $ \mathbf{x} = (x_{i})_{i \in \mathbb{N}} $ is $ \displaystyle \left( \sum_{i=1}^{\infty} |x_{i}|^{p} \right)^{1/p} $. The $ ...
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1answer
60 views

proof that this is an isometric map (on a $C^*$-module)

Are my steps right? I'm not sure about the statement in bold below. Let $A$ be a $C^*$-algebra. Let $X$ be an $A$-module. Let $x\in X$, let $a= \langle x,x \rangle $ Define $\lambda _a (z) = az$, ...
8
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238 views

Proof that the set of doubly-stochastic matrices forms a convex polytope?

Does the set of all doubly-stochastic matrices form a convex polytope? In general, I wonder how the proofs of convexity and geometry can be established for sets of matrices of this kind? Anything to ...
3
votes
1answer
141 views

Computing the Euler Lagrange equations

Let $F(u) = \int_0^1(u'')^2+u^2dx $ on $C^2[0,1]$ satisfying $u(0)=a,u(1)=b,u'(0)=c,u'(1)=d$ where $a,b,c,d \in \mathbb{R}$. If $u_*$ is a minimizer, for $\phi \in C^2[0,1],\ \frac{d}{ds}| _{s=0} ...
2
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1answer
54 views

Separation of function

When can a function of 2 variables say $h(x,y)$ can be written as $$\sum_i f_i(x)g_i(y)$$ I want to know what conditions on $h$ would ensure this kind of separation.
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91 views

Determine whether the terms of the functional are convex or strictly convex

Suppouse $F(u) = \int^1_0 ((u')^2+3u^4+cosh(u)+(x^3-x)(u))dx \:in\: C^1[0,1]$. Where $u(0)=u(1)=0$ Consider the functional term by term. Decide for each term whether it is convex or strictly convex ...
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2answers
354 views

The space of Riemann integrable functions with $L^2$ inner product is not complete

I am trying to find a sequence of Riemann integrable functions on $[0,1]$ converging in $L^2$ to a Lebesgue but not Riemann integrable function. I tried Dirichlet function but could not find a ...
3
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1answer
153 views

Close linearly independent sequence

X is an infinite-dimensional normed space. Show that there is a linearly independent sequence ${x_n}$ in X, such that for any sequence ${\epsilon_n} > 0 $ for all n, there is a sequence ${y_n}$ ...
3
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1answer
77 views

Find adjoint operator

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal basis $(e_n)$. Denote by $V$ the subspace of finite linear combinations of the basis-vectors. Define $T$ on $\mathcal{H}$ with $D(T) = ...
2
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1answer
108 views

Weak and strong convergence of sequence of linear functionals

Is this sequence of linear functionals weakly (strongly) convergent : $$f_n((x_j))=\sum_{k=1}^{n}{\frac{x_k}{k}} , (x_j) \in \ell_2\,?$$
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0answers
75 views

Continuity and openness of the map $C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$

I need to prove or disprove that the composition operator is continuous and open. Consider the following map $$h:C([0,1],[0,1]) \times C([0,1],[0,1]) \to C([0,1],[0,1])$$ that takes a function ...
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0answers
70 views

Fourier transform of integral kernel of the free resolvent

The free resolvent in $\mathbb{R}^3$ has this rapresentation $$(R_0(z)f)(x)=\int_{\mathbb{R}^3}\frac{e^{i\sqrt{z}|x-y|}}{4\pi|x-y|}f(y)dy$$with $\Im \sqrt{z}>0$. Then its integral kernel is ...
1
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1answer
71 views

If $f=\chi_{\{0\}}$, how to show $\|f\|_\infty=1$?

$(X,\Omega,\mu)$ where $X=[0,1],\Omega=$ Borel subsets of $[0,1]$, and $\mu(S)=\infty$ if $0\in S$, otherwise $\mu$ is Lebesgue measure. If $f=\chi_{\{0\}}$, how to show $\|f\|_\infty=1$? Any hint is ...
4
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1answer
215 views

Polynomial root (using contraction mapping principle)

I am asked to provide an iterative algorithm which would lead to finding a real root of this polynomial: $$6x^5-x^3+6x-6=0$$ It is required to rely on the contraction mapping principle and Banach ...
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2answers
58 views

closest approximation to finite dimensional subspace

I want to prove that if $X$ is a normed space, $A$ is some finite dimensional subspace of $X$, and $x \in X/A$, then there is some $y\in A$ such that $d(x,A) = \|x - y\|$. I am familiar with this ...
1
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1answer
102 views

How can I study the continuty of this function?

Let $f\in L^2(\mathbb{R}^3)$ with compact support; is the function $$F(z)=\int_{\mathbb{R}^3}dx\bigg(f(x)\frac{e^{i\sqrt{z}|x|}}{4\pi|x|}\bigg)$$ continuous in the set $$Q=\lbrace{z: \Re z\in [a,b], ...
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121 views

Are these sets compact? [closed]

I've some problems concerning this question: Are the following sets compact in $C_{[0, 1]}$ where $ d(x(t), y(t))=\sup_{[0,1]}|x(t)-y(t)|$: $${\{x(t) \mid x(t)=e^{t-a}, a>0\}},~~{ \{ x(t) ...
12
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1answer
400 views

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for ...
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2answers
442 views

Separable Hilbert space have a countable orthonormal basis

I want to show that every an infinite-dimensional separable (contains countable dense set) Hilbert space has a countable orthonormal basis. I know that every orthogonal set in a separable Hilbert ...
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1answer
64 views

Isomorophisms on $Lp$ and on $lp$?

I want to prove that $lp$ is isomorphic to the infinite dierct sum of $lp$, similarly for $Lp$. Every time I try to define an operator, I lose one of the properties that this operators must have like ...
2
votes
1answer
208 views

Norm of element of Hilbert space as supremum over dense subspace?

Suppose $H_1 \subset H_2$ are both Hilbert spaces with different inner products $(\cdot,\cdot)_{H_1}$ and $(\cdot,\cdot)_{H_2}$. Suppose also that $H_1$ is dense in $H_2$ and that the inclusion is ...
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2answers
104 views

What should I know about Sobolev space?

I have done some Sobolev spaces with some embedding theorems, trace theorems etc. Sorry that my question is really vague. If my professor asks me what is great about Sobolev space, what should I ...
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0answers
37 views

pseudoinverse under change of norm

Let $X$ be a Hilbert space. Let $T : X \rightarrow X$ be a linear mapping. Suppose we have two scalar products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ on $X$. Let $T_1$ and ...
4
votes
2answers
493 views

About Baire's Category Theorem(BCT)

Consider the following theorem known as Baire's Category Theorem (BCT). Theorem.[BCT] A non-empty complete metric space $X$ is not a countable union of nowhere dense sets. I am interested on how to ...
3
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2answers
68 views

Some space of sequences is complete.

Let A be a Banach space and B be the space of all sequences in A that converge to 0, with norm being the maximum element of the sequence. I need to prove that B is complete. What I've done so far: ...
3
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1answer
165 views

Inequality concerning a Holder continuous function composed with a diffeomorphism

I'm trying to fill in the details for the following inequality from a paper, but am thoroughly stumped. Prelude Let $f \in C_c^{\gamma}(\mathbb{R}^n)$ for some $\gamma \in (0,1)$ (that is, a ...
2
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1answer
351 views

Hölder's inequality different condition for equality ($L^p$ space)

I came across this condition for equality for Hölder's of which I would like to know proof of. Equality holds if and only if $A \vert f \vert^{p-1} = B \vert g \vert$ for non negative $A,B$ not both ...
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1answer
81 views

Local Convexity and norms

Any help with this doubt is appreciated. I would like to understand the relation between local convexity and norms. I know that all norms are locally convex. But let us say for example I want to ...
2
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0answers
72 views

Sobolev spaces in $\mathbb R^n$ with functions having support on a closed set

I am interested in $H^s$ Sobolev spaces in $\mathbb R^n$ which have functions with support in a given closed set $K$ , denoted by $H^s_K$. Here $K$ is the complement of some bounded open set $\Omega$. ...
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2answers
114 views

Norm of element in Hilbert space: can I take supremum over subspace like this?

Let $X \subset Y$ be Hilbert spaces (with different norms), with continuous and dense injections. We have $$\lVert y \rVert_Y = \sup_{v \in Y, \lVert v \rVert_Y \leq 1} |(y,v)_Y|$$ Is it true that ...
4
votes
1answer
35 views

Banach contraction theorem for partially defined maps

Let $(X,d)$ be a metric space and let $(f,D(f))$ be a partially defined map on $X$, i.e. $D(f)\subset X$ and $$ f:D(f)\to X. $$ Suppose that $f$ is a contraction, i.e. $$ d(f(x),f(y))\leq ...
2
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0answers
85 views

Limit in norm of a Sobolev space

I consider, for $s>\frac{1}{2}$, the space $L^{2,-s}(\mathbb{R}^3)=\bigg\{f: \int_{\mathbb{R}^3}|f(x)|^2(1+|x|^2)^{-s}<\infty\bigg\}$ and I have to show that the function ...
0
votes
1answer
41 views

$ T = V|T| $ in a finite-dimensional Hilbert space.

Let $ T \in B(H) $, where $ H $ is a finite-dimensional Hilbert space. Is it true that $ T = V|T| $ for some unitary operator $ V $?
2
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0answers
134 views

Compact injections and equivalent seminorms

Let $V$ and $H$ be two Banach spaces with norm $\lVert \cdot \rVert$ and $\lvert \cdot \rvert$ respectively such that $V$ embeds compactly into $H$. Let $p$ be a seminorm on $V$ such that $p(u) + ...
4
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0answers
55 views

Fredholm alternative for polynomially compact operators

Let $T \in \mathcal{L}(E)$ be a polynomially compact operator, i.e., there exists a polynomial $p$ such that $p(T)$ is a compact operator. Suppose $p(1) \neq 0$. I want to show that $N(I-T) = \{0\} ...
3
votes
1answer
92 views

Functional Analysis Proof

I'm sorry that the title is very general indeed. I'm looking for a theorem/corollary that uses all of the following four theorems/concepts in its course. This may be rather ambitious, but any ideas? I ...
4
votes
0answers
89 views

Open map in Banach algebra

I'm having trouble showing a certian function is open and can be extended. Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded ...
0
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1answer
38 views

Identifying 2 spaces of distributions

Hoi, I want to show that the space $C^{\infty}(\Omega)'$ of continuous linear functionals on $C^{\infty}(\Omega)$ can be identified with the subspace $\mathcal{E}'(\Omega)$ (distributions with ...