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, ...

learn more… | top users | synonyms (1)

1
vote
1answer
14 views

A Banach space in between $L^{1}$ and $L^{2}$, does it make sense?

Let $L^{p} (A, B)$ be a collection of functions $f:A \mapsto B$ satisfying $$(\|f\|_{p})^{p} := \int_{A} |f(x)|^{p} dx <\infty.$$ Now we consider functions $f:\mathbb R^{2} \mapsto \mathbb R$. We ...
1
vote
0answers
16 views

$L^2$-Sobolev space

I am looking at the proof of the following lemma and I don't understand the conclusion. Lemma: For $k\geq 0$ integer, let $f=\sum_{n\in \mathbb{Z^d}}{c_n\chi_n}\in L^2(\mathbb{T}^d)$. Then $$f\in ...
3
votes
1answer
36 views

Non-compactness of the resolvent

Consider a complete non-compact Riemannian manifold $M$ and the resolvent of the Laplacian $(-\Delta + \lambda I)^{-1}$. It is known that the resolvent is in general not a compact operator. I am ...
2
votes
1answer
31 views

Borel regular measure: Approximate any measureable set by compact sets

Let $(K,\mathcal{F},\mu)$ be a measure space. Let $K$ be a compact Hausdorff space and $\mu$ be a regular finite measure. We said that it is regular if $\mu(A) = \inf\{\mu(B): B \text{ open }, ...
2
votes
2answers
17 views

Convex cone of nonnegative functions in L2 has empty interior

Convex cone $S:=\{f\in L^2(\mathbb{R},\mu):f\geq 0\}$ has empty interior in $L^2(\mathbb{R},\mu)$ when $\mu$ is Lebesgue measure. I wanted to prove it but i have major holes in my knowledge of ...
3
votes
0answers
22 views

When can we exchange the trace and an integral/limit/derivative?

For a trace class operator $A$ (acting on a Hilbert space), that is parameterised by a real variable $x$, what are the conditions for the following to hold? $$ \mathrm{tr} \int_a^b A(x) \, dx = ...
0
votes
4answers
66 views

Is an non-invertable matrix an linear operator?

I am under the impression that any matrix can be called a linear operator, even if the matrix does not have an inverse. Is it true? There are many properties a linear operator enjoys; do all matrices ...
0
votes
0answers
14 views

Two-dimensional subespace suplementary of another one

Let E be a real normed space. All subespace S of codimension 1 (hyperplane) in E is either, closed or dense. What do say about a similar property when S is of codimension 2?
-1
votes
1answer
13 views

symetric closed operator and extension [on hold]

i have this question let A a symetric closed operator let pose that A have a self adjoint extension is possible that A has an extension such that closure A can't have a self adjoint extension
1
vote
1answer
18 views

Completely positive maps

Let $B$ be a commutative C$^*$-algebra and let $M_n$ denote the algebra of $n\times n$ complex matrices. Let $f$ be a state on the tensor product of $B$ and $M_n$, $B\otimes M_n$. How can I show that ...
3
votes
2answers
42 views

An infinite dimensional normed linear space is the union of two disjoint convex sets

Let $X$ be an infinite dimensional normed linear space. I want to show that there exist two disjoint convex sets $C_1$ and $C_2$ such that $X=C_1\cup C_2$ and both $C_1$ and $C_2$ are dense in $X$. I ...
1
vote
1answer
37 views

Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
0
votes
0answers
28 views

a question on Frechet derivative

Suppose the derivative of a functional is given by $\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi$ for $\phi\in W_0^{1,p}(\Omega)$, then what is the functional?.
0
votes
1answer
30 views

If $u \in H^1(0,\infty)$, does $u_x(\infty) = 0$?

Let $u \in H^1(0,\infty)$. Then is it true that $u'(x) \to 0$ as $x \to \infty$? I am wondering by Green's formula/IBP in this setting; do I get something like $$\int_0^\infty u_{xx}v = ...
0
votes
0answers
12 views

Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
0
votes
1answer
21 views

Is $B_{\ell_1}$ weak-metrizable?

I know that for a Banach space $X$, the unit ball $B_{X}$ is weak metrizable if and only if $X^*$ is separable. My question is that Is $B_{\ell_1}$ weak-metrizable?
4
votes
1answer
40 views

Isolated Eigenvalue

What does it mean that an eigenvalue is "isolated"? My intuitive understanding says it is when one can find an open ball around it such that there is no other eigenvalue in that open ball. However, I ...
3
votes
1answer
30 views

Intuition of weak star convergence.

Given $\Omega=(0,1)$, consider the following sequence $$ v_j(x)\colon=\begin{cases} \;a &\text{if }jx-\lfloor jx \rfloor\le\theta\\ \;b &\text{otherwise} \end{cases} $$ where ...
2
votes
0answers
52 views

Functional Maximization

So how do we solve a problem like this: Find the function $s(x)$ such that $s(x)$ maximizes $$\int_0^{s^{-1}(k)} s(x) dx $$ where $x\in[0,10]$, $s(x)\in[0,1]$, and $k\in[0,1]$ ($k$ is a constant). ...
1
vote
1answer
27 views

Eigenfunctions of $-\Delta_{S^n}$

I read somewhere that the eigenvalues of the Laplacian $-\Delta_{S^n}$ on the sphere $S^n$ consist of $k^2 + (n - 1)k$, with the corresponding eigenspace $V_k$ consisting of homogeneous harmonic ...
2
votes
1answer
22 views

Existence of Certain Locally Integrable Function Defining a Tempered Distribution

We say that a distribution $T$ is tempered if for every sequence $\left\{\phi_{n}\right\}$ in $C_{c}^{\infty}(\mathbb{R}^{n})$ tending to $0$ in the topology of the Schwartz space ...
1
vote
2answers
25 views

Show that properties of norm are satisfied

Show that \begin{align} & \|y\|_M= \max_{a \leq x \leq b} |y(x)| \tag 1 \\[8pt] & \|y\|_1=\int_a^b |y(x)|\, dx \tag 2 \end{align} satify the properties of a norm in $C[a,b]$. That's what I ...
0
votes
0answers
20 views

How do I compute this metric projection?

I saw a result that says: Given a nonzero vector $a$ and the convex set $K:=\{y\in H: \langle a,y\rangle =\alpha, \alpha \in \mathbb{R}\}$ a hyperplane, then $$P_Kx=x-\frac{\langle ...
0
votes
1answer
32 views

Is there a relation between Ill-posed problems and Eigenvectors.

One can easily explain an ill-posed problem with an equation AX=b. The following link is an good example: http://www.encyclopediaofmath.org/index.php/Ill-posed_problems 1) Can there be a class of ...
0
votes
2answers
60 views

Is $[-1,1]$ complete under the Euclidean metric? [on hold]

Is it true that the interval $[-1,1]$ is complete under the Euclidean metric?
2
votes
1answer
27 views

Normed space where unit ball's weak and norm topology coincide?

Let $X$ be a normed space. Are there any infinite-dimensional examples such that the $(B_X, w\restriction_B) = (B_X, \|\cdot \|\restriction_B)$. In finite dimensions this obviously holds always true. ...
-2
votes
0answers
8 views

under what conditions, a function on $CB(X)$, space of closed and bounded sets, [on hold]

under what conditions, a function on $CB(X)$, space of closed and bounded sets, will be continuous. where function $F$ defined on $CB(X)$ as F(A)={\bigcup F(x):x∈A}
0
votes
0answers
6 views

Deriving information about asymptotics from finitness of a limit

Let $f_1,f_2:\mathbb{R}\setminus\{0\}\to \mathbb{R^+}$ be two $C^1$ functions and $\alpha:\mathbb{R}\setminus \{0\}\to \mathbb{R}$ be a function from a Zygmund class (in particular it is Holder for ...
0
votes
1answer
18 views

Exponential decay estimate involving $C_{0}$-semigroup and the principle eigenvalue

Consider the heat equation defined as $$\begin{cases} \partial_{t}u(x,t)=\Delta_{x}u(x,t) &\mbox{ } t>0\mbox{, } u\in U \\ u(x,t)=0 &\mbox{ } t\ge 0\mbox{, } x\in\partial U \\ u(x,0)=g(x) ...
3
votes
3answers
39 views

Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$?

Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$? My thinking is: $\langle u+v+w,v \rangle =0 \iff \langle u,v \rangle + \langle w,v \rangle = -1$ How do i ...
2
votes
0answers
22 views

Multiplication operators are sectorial

Consider the multiplication operator $M_a$ on $L^p(M)$, where $M$ is a Riemannian manifold, and $a$ is a non-negative function. An operator $A$ is said to be sectorial if there exists $\theta \in (0, ...
0
votes
0answers
15 views

function defined as integral of borel function

I know that $f \in B_b(E)$, where $B_b(E)$ is the set of Borel bounded function on an euclidean space E. I have to show that: \begin{equation} x \to \int_{0}^{+\infty} e^{-at} P_tf(x) dt ...
0
votes
0answers
20 views

For $z=(c_n)\in l^3$ and $N(z)=\left(\sum_{n=1}^{\infty} \left|\frac{c_n}{n}\right|^3\right)^{\frac{1}{3}}$. Show $N(z_1+z_2)\leq N(z_1)+N(z_2)$?

For $z=(c_n)\in l^3$ and $N(z)=\left(\sum_{n=1}^{\infty} \left|\frac{c_n}{n}\right|^3\right)^{\frac{1}{3}}$. Show $N(z_1+z_2)\leq N(z_1)+N(z_2)$? Obviously $$N(z_1+z_2)=\left(\sum_{n=1}^{\infty} ...
0
votes
0answers
21 views

Let $H$ be a Hilbert Space with$\langle \cdot,\cdot \rangle$ and $E_1=\{w\in H : Pw=w\}$, show $E_1$ is closed.

Let $H$ be a Hilbert Space with $\langle \cdot,\cdot \rangle$ and $E_1=\{w\in H : Pw=w\}$ with $P:H\rightarrow H$ is linear, $P^2=P$ and $\langle Px,y \rangle=\langle x,Py \rangle \forall x,y\in H$. ...
-6
votes
0answers
18 views

Is there any apps for downloading study materials of our mathematics papers for exam preparation [on hold]

I need apps details for downloading study materials for exam preparation
0
votes
1answer
12 views

Why does a Hermitian operator with singleton spectrum have to be scalar?

One proof of Schur's lemma proceeds by showing that a Hermitian intertwining operator of an irreducible representation (of a topological group on a Hilbert space) has a spectrum that contains only one ...
3
votes
0answers
21 views

Riesz Lemma with $\alpha=1$ and Linear Bounded Functional

Show that on a normed linear space $X$, Riesz lemma with $\alpha=1 $ holds implies that every bounded linear functional attains its norm on the unit sphere of $X$. This is not a homework question and ...
0
votes
1answer
32 views

Show if $x\in l^p$ and $y\in l^q$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ then $\|xy\|_r\leq\|x\|_p\|y\|_q$

Show if $x\in l^p$ and $y\in l^q$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ then $\|xy\|_r\leq\|x\|_p\|y\|_q$. My intuition is to use Young's Inequality and then apply it to $A_k=\frac{|x_k|}{\|x\|}$ ...
-1
votes
0answers
20 views

If F is upper semi continuous closed set- valued map

If $F$ is upper semi continuous closed set-valued map from $X$ to $Y$ and $A⊂X$ and $X$ be a complete metric space. The image of the set $A$ under $F$ is given by $$F(A) = \lbrace y: y \in F(x): x ...
1
vote
0answers
10 views

If F is a closed set- valued map from X to Y and A is a subset of X.

If $F$ is a closed set- valued map from $X$ to $Y$ and $A \subset X$, then the image of the set $A$ under $F$ is given by $F(A) = \lbrace y: y \in F(x): x \in A \rbrace $, let $CB(X)$ denote space ...
0
votes
0answers
15 views

If a mapping on complete metric space is continuous, then mapping on space of closed and bounded sets of that metric space is also continuous ??

If a mapping on complete metric space $X$ is continuous, then mapping defined for sets, on space of closed and bounded sets of $X$ is also continuous ??
2
votes
0answers
22 views

Resolvent estimate of hyperbolic Laplacian

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda ...
1
vote
1answer
23 views

Can we conclude that $v_{n}\rightarrow v$ in $L^{\infty}\left(\Omega\right)$ if $p>N$

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $v_{n}\rightharpoonup v$ in $W_{0}^{1,p}\left(\Omega\right)$ , $\left\Vert v_{n}\right\Vert _{W_{0}^{1,p}}=1$ $\forall n$ . So we ...
-1
votes
0answers
18 views

an objective question from functional analysis [on hold]

Let $A$ and $B$ be bounded operators on a Hilbert space $H$ such that $AB=BA$. Let $\lambda$ be an eigenvalue for $A$. Then it must be that a)$B$ has no eigenvalue b)$B$ has at least one ...
2
votes
1answer
29 views

Showing that an operator generates a contraction semigroup

Let $A$ be the infinitesimal generator of a contraction semigroup $(T(t))_{t\ge 0}$ on the Hilbert space $X$, and $D\in\mathcal{L}(X)$. I want to show that the operator $A+D-2\|D\|I$ with domain ...
1
vote
0answers
20 views

Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$, then ...
6
votes
2answers
28 views

Prove Operator is a Projector

Let $\mathscr{H}$ be a complex Hilbert space. A projector is a linear map $P:\mathscr{H}\to\mathscr{H}$ such that $P\circ P = P$. I'm trying to prove the following claim, from the information given ...
1
vote
0answers
18 views

Continuity of Translation and Dilation on $L^p$ spaces

Let us consider any $f \in L^p(U)$, where $U \subset \mathbb R^n$ is open, and $1 < p < \infty$. We know the translation operator $f(x) \mapsto f(x+a)$ and the dilation operator $f(x) \mapsto ...
0
votes
0answers
32 views

Integral equation with exponential

I would like to solve the following integral equation for $u(t)$, where $\theta, \gamma, \lambda, \kappa$ and $\sigma$ are parameters, but I haven't managed to obtain a solution so far. Any hints on ...
2
votes
1answer
46 views

Calculating the distance function and projection

Let $\Omega=\{x\in\mathbb R^n\mid\langle a,x\rangle=b\}$. We define the distance function and projection as follows $$d(x;\Omega)=\inf\{||x-\omega||\mid\omega\in\Omega\}$$ ...