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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1answer
38 views

Well posedness and regularity of diffusion advection with Robin BC

I have the following diffusion advection with time and space dependent coefficients with Robin BC \begin{equation} \left \{ \begin{array}{l} \partial_t u - div(B(t,x) \nabla u) + V(t,x) \nabla u = 0 ...
1
vote
1answer
60 views

Fabulous Sobolev inequality

Today I was told by my tutor that on closed manifolds like $\mathbb{S}^2$ (we can stick to $\mathbb{S}^2$ here) the following Sobolev inequality holds (if the right-hand side exists, the left-hand ...
0
votes
0answers
52 views

Problem with applying Lagrange (mean value theorem) in a proof.

Suppose $A \in R^n$, $A$ open, and we have a function $f:A \rightarrow R^m$ Theorem: If $f \in C^1_A$ then $f$ is differentiable in every point of $A$. where $f \in C^1_A$ if $f_j$ has partial ...
1
vote
1answer
54 views

Dual Cone Containment

I have the following conjecture which, as of now, I can neither prove nor disprove: Let $H$ be a hilbert space, and let $C \subset H$ be a closed (convex) cone with the property that ...
4
votes
2answers
101 views

Operator norm of positive operator.

I'm studying Reed and Simon's "Methods of Modern Mathematical Physics" Vol. 1 (http://www.math.bme.hu/~balint/oktatas/fun/notes/Reed_Simon_Vol1.pdf). In the proof of the square root lemma (p.196) they ...
1
vote
1answer
18 views

One step in proving Kakutani-Krein Theorem (about the closed lattice separating points)

This question is about a step in Reed and Simon's functional analysis book, in proving Kakutani-Krein Theorem. (Any closed lattice $\mathcal{L}$ in $C_\mathbb{R}(X)$, where X is compact Hausdorff, ...
10
votes
1answer
93 views

If $f\in C[0,1)$, $\int_{0}^{1}f^{2}(t)dt =\infty$, can one construct $g\in C[0,1)$ so $\int_{0}^{1}g^{2}dt < \infty$, $\int fgdt = \infty$?

If a real $f\in C[0,1)$ satisfies $\int_{0}^{1}f^{2}(t)dt =\infty$, can one explicitly construct $g\in C[0,1)$ such that $\int_{0}^{1}g^{2}dt < \infty$ and $\int_0^1 fgdt = \infty$? There must ...
0
votes
1answer
20 views

A inequality concerns with eigenvalue of a symmetric and positive definite matrix

How to prove the statement: Let $A$ be a symmetric and positive definite matrix of order $n$. Suppose that $\lambda_{\min}\ge \lambda_0>0$, where $\lambda_{\min}$ is the smallest eigenvalue of $A$. ...
1
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2answers
30 views

Definition of “Extension” of Bounded Linear Transformation

I have been given the problem of proving the B.L.T. Theorem for my homework which states, Every bounded linear transformation $\mathsf{T}$ from a normed vector space X to a complete, normed vector ...
2
votes
1answer
29 views

A restriction of a symmetric operator such that the range of (operator)+i is the same

I have this problem and I really can't see how to do it. Suppose that $C$ is a symmetric operator, $A\subset C$ and that $\operatorname{Ran}(C+i)=\operatorname{Ran}(A+i)$. Prove that $C=A$. ...
0
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1answer
26 views

The condition $\lambda({x; |f(x)|>a})< \infty$ for any $a>0$.

In class we discussed the Sobolev inequality and in its statement we had that some function $f$ has to satisfy $\lambda( \{ x; |f(x)|>a \})< \infty$ for any $a>0$, where $\lambda$ is the ...
1
vote
1answer
44 views

Is the $C[-1, 1]$ complete in $||f|| = \sqrt{\int_{-1}^{1}{f^{2}(t) dt}}$

I'm considering the following problem: For a given space $H = C[-1, 1]$ with inner product $(f, g) = \int_{-1}^{1}{f(t) g(t) dt}$ i'm considering a subspace $H_{0} = \{f \in H | \int_{-1}^{0}{f(t) dt ...
3
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2answers
66 views

A linear function on the space $c_{00}$ that is not continuous

Consider the space of eventually zero sequences: $$c_{00} = \left\{ x = (x^{(1)},x^{(2)},\dots,x^{(k)},\dots)\in\ell^\infty \,\middle|\, \exists k_0 \text{ such that $x^{(k)}=0$ for ...
2
votes
1answer
23 views

Quotient topology of a topological vector space is translation-invariant

Let $(L,\tau)$ be a topological vector space over $\Bbb{C}$ and $M$ be a subspace of $L$ and let $$f:L\to L/M$$ be the canonical map of $L$ onto $L/M$.Let $ \hat \tau$ be the quotient topology on ...
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votes
2answers
73 views

Consider the following map:

$T:(C^1[0,1],||.||_u)\to\ (C[0,1],||.||_u)$ $T(f)=f'$ it is obvious that T is a linear operator.since$(cf)'=cf'$. Consider $f_n(t)=t^n, t \in [0,1]$ Here,$||t^n|| \le1$. Also $||f_n||=1,\ \forall n ...
1
vote
1answer
33 views

prove that $\ell_2$ Parallelepiped is compact (problem in notations for Nested sub-sequences!)

Let $A=\{(x_n)_{n\in\mathbb{N}}\;\big| \;|x_n|\leq\frac1n\quad ;n=1,2,\cdots\}$ be a subset of $\ell_2$, which is a Parallelepiped in $\ell_2$. I think I've stuck in a problem with notation ! ...
2
votes
0answers
73 views

Image of a precompact under the action of uniformly continuous function is a precompact

Suppose we have two metric spaces $(X, \rho_x)$ and $(Y, \rho_y)$ and a uniformly continuous function $f\colon X \to Y$. The problem is to prove that image $f(A)$ of every precompact $A \subset X$ ...
0
votes
1answer
26 views

Equivalent norms proof

Is the standard $L^2$ norm $$\|u\|^2=(u,u)$$ equivalent to a weighted $L^2$ norm $$\|u\|^2_g =(gu,u)$$ with g>0? If so, how can one prove this?
2
votes
1answer
28 views

integral of the square laplacian

I am reading p.380 Evans PDE 2nd edition, on the derivation of estimates. I am looking for the proof of the following equality $$\int_{\mathbb{R}^n}(\Delta u)^2 dx=\int_{\mathbb{R}^n}|D^2 u|^2 dx$$ ...
1
vote
1answer
29 views

Basic neighborhoods in weak topology

I am trying to visualize the basic neighborhoods of the form $V(x_0;\varepsilon,f_1,...,f_n) = \bigcap_{j=1}^n \{ x \in E : |f_j(x-x_0)|<\varepsilon \}$ where $x_0 \in E$, $\varepsilon>0$ and ...
4
votes
1answer
75 views

Prove that $f(\lambda,v)=\lambda v$ is continuous

Let $V$ be a vector space with a metric $d$. Suppose that this metric satisfies: (1) $d(v+z,w+z)=d(v,w)$ $\forall v,w,z\in V$ (2) $d(\lambda v,\lambda w)=|\lambda|d(v,w)$ $\forall v\in V, ...
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votes
1answer
22 views

Is this condition on linear functionals equivalent to boundedness?

Let $T : C_c(X) \rightarrow \mathbb{R}$ be a linear functional satisfying $$\sup \{|T(f)| :\ f \in C_c(X), |f| \leq 1, \text{Supp}(f) \subset K\} < \infty$$ for each compact $K \subset X$. Is ...
2
votes
1answer
20 views

Existence of operator with certain properties on a Banach space

I ran across this question, and was a little puzzled by it. I neither know how to solve it, nor its meaning: Let $X$ be a Banach space, and let $A,B$ be bounded linear operators on $X$ such that $A$ ...
1
vote
1answer
14 views

Abel transform a topological isomorphism?

Given a noncompact semisimple Lie group $G=NAK$ with Weyl group $W$, consider the symmetric space $X=G/K$. Let $f$ be a function in $D(X)^K$, the space of $K$-invariant functions on $X$. Then the Abel ...
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0answers
28 views

Reference Request: Evaluation Relative Bound 0 to Differentiation

In Kato, Perturbation Theory of Linear Operators, Chapter 4, Section 1, Example 1.8, pp. 192-193, it is shown that for all absolutely continuous functions $u: (a, b) \to \mathbb{C}$ , $a < b$ ...
1
vote
1answer
28 views

Notion of a limit for a sequence of matrices growing in both dimensions

Suppose $\{M_{N_1,N_2}\}$ is a sequence of $N_1 \times N_2$ matrices whose elements are non-negative and are bounded from above by $1$. I wonder if there is a way to define a notion of convergence and ...
2
votes
1answer
55 views

Discontinuous bilinear form separately continuous

Do you have an example of a real normed space $V$ and a bilinear form $B : V \times V \to \mathbb R$ that is discontinuous but such that $B$ is separately continuous for each variable? $V$ has to be ...
2
votes
1answer
32 views

Convergence in a sequence space (again)

Apologies for another possibly easy question on this topic... I am just trying to learn sequence spaces and get some intuition so I keep thinking about various examples. Consider an infinite ...
1
vote
1answer
63 views

Convergence in a sequence space

For any $n\in \mathbb{N}$ consider the following sequence: $$a_n=\left(\underbrace{\frac{1}{n}, \ldots, \frac{1}{n}}_{n \text{ elements}}, 0, 0, 0, 0, \ldots \right).$$ Is there a way to introduce a ...
0
votes
0answers
50 views

To show Banach Operator Norm's equivalence

I am thinking the sentence which may be about the Banach algebra function: Let $A \in \mathcal{B}(H)$ where $H$ is Hilbert. \begin{equation} \| A \| = \sup_{ \| u \|, \| v \| \leq 1} | \langle Au, ...
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0answers
18 views

The function $u\rightarrow \int_A|\nabla u|^2dx$ is continous

Let be $A$ an open set in $\mathbb{R^N}$ and $u\in C^1(A,\mathbb{R^N})$, How can I show that the function $$u\rightarrow \int_A|\nabla u|^2dx $$ is continous? Thanks
2
votes
2answers
24 views

Example of an operator whose spectrum satisfies this condition

Give an example of a Hilbert space $\mathcal{H}$ and an operator $A: \mathcal{H} \to \mathcal{H}$ satisfying $$\sigma \left( A \right) = \varnothing \neq \sigma \left( A^2 \right),$$ where $\sigma ...
3
votes
2answers
77 views

Proving that a linear operator $T$ is continuous.

If the space $X$ is banach , then I want to show that any linear map $T:X \to X$ is continuous iff the null space is closed. I could show that if $T$ is continuous then the null space is closed. But I ...
2
votes
2answers
52 views

uniqueness of the solution of heat equation in convolution form

I wish show the solution expressed in the form $$u(t,x)=\int_{\mathbb{R}}\Phi_t(x-y)f(y)\,dy$$ is unique for any $f\in\mathcal{S}(\mathbb{R})$ (the space of rapidly decreasing functions in ...
3
votes
1answer
92 views

What is predual space to Radon measures with finite moment?

Define for any $p\in\mathbb{N}$ space $\mathcal{M}^p(\mathbb{R}_+)$ as measures with finite $p -$moment, i.e. such Radon measures $\mu$ that $\int_0^\infty x^p d\mu(x) <C_{p,\mu}$. What is the ...
0
votes
1answer
33 views

Coordinate projections in $l^p$ are Lipschitz continuous

Prove that for all $p\in[1,\infty]$ the k-th projection $\pi_k:l_p\to \mathbb R$ $\pi_k(x)=x_k$ is Lipschitz continuous I need to prove that for all $x,y\in l_p$ $\exists K>0$ such that ...
1
vote
2answers
57 views

How can I find an interval where $f(x)=\frac 12(x+\frac 3x)$ is contractive mapping?

I want to find out an interval where $f(x)=\frac12(x+\frac 3x)$ is contractive mapping. How can I find this interval where $f(x)$ becomes contractive?
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votes
0answers
57 views

(Exponential) Growth of Operator Norm of Uncentered Maximal Function

Define the uncentered Hardy-Littlewood maximal operator $M$ by $$Mf(x):=\sup_{x\in B}\dfrac{1}{\left|B\right|}\int_{B}\left|f\right|,$$ where we the supremum is taken over all (open) balls $B$ ...
3
votes
0answers
60 views

Outline of Generic Separable Banach Spaces don't have a Schauder Basis

So, I know P. Enflo showed that there is a separable Banach Space that doesn't satisfy the approximation property. My professor mentioned during class that in fact generic separable Banach Spaces ...
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vote
3answers
65 views

Riesz Representation Theorem proof in Bjork

I have a question regarding the proof of Riesz representation Theorem in the appendix of Bjork textbook. Here is the statement of the theorem: Assume that $F:H\to R$ is a bounded linear functional. ...
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0answers
19 views

About calculating limits of integrations (Part 3)

Let $L,a,x,y,t$ be positive numbers. Where $t$ is the only variable and the rest are constants. Let $z_1 = i(a+x)-y-it$ Let $z_2 = i(a+x)+y-it$ Now look at these two integrals, $\lim_{x,y ...
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1answer
54 views

About calculating limits of integrals (Part 2)

The function $tanh(\pi z)$ has its poles at the points $i(n+\frac{1}{2})$ for $n \in \mathbb{Z}$. Now I want to take an $\epsilon$ circle around such a pole and contour integrate the function $z\text{ ...
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votes
0answers
18 views

Notation about integration with respect to abstract Borel measure.

For the Borel measures $\mu \in \mathcal{M}(\mathbb{R}^N, \mathbb{R}^M)$, is it natural to define an abstract integral for $\mathbb{R}^M$ in general? Since my book doesn't mention this at ...
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2answers
51 views

Consider the Hilbert product space $X\times X$

Consider the Hilbert product space $X\times X$. In $X\times X$ define the closed convex 'diagonal' set by $$D={(x,x):x\in X}$$ Obtain a formula for projection $P_D$ and rigorously prove it. I really ...
0
votes
2answers
29 views

Differentiating between boundedness and finiteness.

I am a little puzzled by some notations in optimization community. Is there anyone who can explain why $f_1:\mathbb{R}^n\rightarrow\mathbb{R}$ is a finite valueed but ...
1
vote
0answers
36 views

Can you explain how function can be smooth but not finite valued?

I was reading Kurdyka-Lojasiewicz property in a paper and the made an assumption that f must be smooth and finite valued. I am wondering if one can you explain how function can be smooth but not ...
0
votes
1answer
98 views

Alternative approach of a step in the proof of the Banach-Alaoglu Theorem

Let $V$ be a normed vector space and denote the closed unit ball in $V$ as $B$ and the closed (with respect to the norm topology) unit ball in $V^*$ as $B^*$. Let $D=\{z\in\Bbb{C}\mid |z|\leq 1\}$. ...
1
vote
1answer
25 views

Is $\liminf_{\epsilon \to 0} \frac{1}{\epsilon}\int_0^T \int_\Omega \nabla F(v)\nabla (T_\epsilon(u-v)) \geq 0$?

With $\Omega$ a bounded domain, let $$A(\epsilon) = \frac{1}{\epsilon}\int_0^T \int_\Omega \nabla F(v)\nabla (T_\epsilon(u-v))$$ where $T_\epsilon$ is the truncation at height $\epsilon$: ...
2
votes
2answers
34 views

Derivative with respect to function

I am looking to calculate the derivative of a functional $\phi(\rho)$ with respect to $\rho$, that looks like $$\phi[\rho](x)=\rho(x)\int_0^1\log|x-y|\rho(y)dy.$$ I have read that the Gateaux ...
1
vote
1answer
44 views

Does the canonical $\pi: X \to X/Y$ map the closed unit ball to the closed unit ball?

Let $Y \subset X$ be a closed subspace of the normed space $X$. Consider $\pi: X \to X/Y, x \mapsto [x]$. Then for $x \in X, ||x||\le 1$: $\quad||[x]|| = \text{inf}_{y \in Y} ||x-y|| \le ...