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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0answers
39 views
Lower bound for an $L_p$ norm of a polynomial
Let $p$ be a polynomial of degree $d$, say.
How to show that
$|p^{(k)}(0)|^r\leq c\int_0^1|p(t)|^rw(t)dt$, for all $k=0,\ldots,d-1$, $r\geq 1$
and some positive w(t) and c?
0
votes
1answer
25 views
Self-adjoint operator on a Hilbert space.
Let $T$ be a self-adjoint operator on a Hilbert space $H$. If for all $x\in H$, $\langle Tx,x\rangle=0$, is $T=0$?
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votes
0answers
23 views
Convexifying Functions
I have the following question:
Imagine you have a function $F(x,y(x),y'(x)), F:\mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} $ and this function is not convex.
Then you can ...
7
votes
1answer
100 views
Proof that there is no Banach-Tarski paradox in $\Bbb R^2$ using finitely additive invariant set functions?
I am wondering if anyone is familiar with the above topic? I have found a proof that it is possible to define a finitely additive invariant set function in $\mathbb{R}^2$ on the circle in Lax's book ...
1
vote
1answer
34 views
Residual spectrum is empty
I'm following Kreyszig's "Introductory Functional Analysis with Applications" and am trying to follow the proof of the following Theorem (9.2-4 on p. 468)
For a bounded self-adjoint linear operator ...
1
vote
0answers
36 views
Compactness in $L^p$
I am studying this article:
http://arxiv.org/pdf/0906.4883.pdf
There is a little part that I do not understand, in the proof of theorem 5, page 4.
Let P be the projection map of $L^p(\mathbb{R}^n)$ ...
0
votes
0answers
24 views
Simple heat equation, solution regularity
I have a small problem with a regularity result for a simple parabolic heat equation:
Given a $C^2$ open subset of $\mathbb R^n$ called $\Omega$, and a time $T > 0$, i have the following heat ...
5
votes
1answer
55 views
Zeta Regularized Determinant of Laplacian
Can anyone point me to a resource where the zeta regularized determinant of the Laplacian is explicitly computed for simple two dimensional surfaces, say a rectangle or torus or cylinder?
2
votes
1answer
31 views
Can I deal with the weak derivative in the “strong” sense?
This is an exercise in functional analysis:
For $k=1,2,3$, let $A_k: D(A_k)\subset L^2([0,1])\to L^2[(0,1)]$ be the first-order differential operators $A_ku=iu'$ with domains
$$
D(A_1) = ...
2
votes
1answer
53 views
About Equicontinuous and Boundedness
Let $X$ be a TVS and $X'$ denotes the space of all continuous linear functionals on $X$. Let us denote the $weak^*$-topology on $X'$ by $\sigma(X',X).$
My question is this. Why does every ...
1
vote
0answers
25 views
Find spectrum of the operator and an explicit form for the solution
Let $A: L^2[0,2\pi] \to L^2[0,2\pi]$ be defined by $Au=v$ if
$v(x)=\int^{2\pi}_{0} cos(x+t)u(t)dt$
(a) Find the point spectrum, eigenspace, residual spectrum, continuous spectrum, and spectrum
(b) ...
1
vote
0answers
32 views
Find spectrum of the operator
I need some help to find point spectrum, residual spectrum, continuous spectrum and for this problem.
Let $0 \in\mathbb{C}$. On the complex space $L^2[-1,1]$ consider the operator
$Au=v,$ where ...
2
votes
1answer
51 views
Second annihilator of subspace is the subspace itself?
Let $X$ be a Banach space over $\mathbb{C}$ with dual space $X'$ and let $M,N$ be subspaces of $X',X$ respectively. Define the annihilator subspaces of $M$ and $N$ as
$$
M_\circ = \{x \in X: f(x) = ...
2
votes
1answer
49 views
When is a subset of $\ell^2$ compact?
I have been looking on the internet for hours now and even asking in chat without an answer. When is a set $M\subseteq\ell^2$ compact? For $L^p$, there is the Arzelà –Ascoli theorem that provides a ...
3
votes
1answer
67 views
Convergence of operator norm
I have a linear bounded operator $A:L_2(0,1) \rightarrow L_2(0,1)$ satisfying $\|A^n\|^{1/n} \rightarrow 0$. Thus, for some sufficiently large $N$, $\|A^N\| < 1$ and then from Gelfand's formula, I ...
2
votes
1answer
35 views
about well-defined integral kernel
Let $\phi:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$ measurable function such that
$$ \int_{\mathbb{R}^n}|\phi(x,y)|\ dx \leq M\ , \quad \int_{\mathbb{R}^n}|\phi(x,y)|\ dy \leq M\,.$$
Let $f\in ...
4
votes
0answers
58 views
Maximal ideals in the algebra of continuously differentiable functions on [0,1]
This is an exercise in Rudin's Functional Analysis, in the chapter on commutative Banach algebras. My (uneducated) guess was that every homomorphism on $C^{1}[0,1]$ is an evaluation at some point of ...
0
votes
1answer
23 views
Is my proof of characterisation of self-adjoint operators on complex Hilbert spaces okay?
I wish to show the following theorem:
Let $T:H\to H$ be
a bounded linear operator on a complex Hilbert
space $H$. Then if $\left\langle Tx,x\right\rangle \in\mathbb{R}$
for all $x\in H$, then $T$ is ...
2
votes
1answer
40 views
About a Weak Topology of a Vector Space
Let $X$ be a real vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each $p\in P$ ...
1
vote
1answer
34 views
About a Weak Topology on TVS(part 2)
Let $(X,\tau)$ be a topological vector space and suppose $P$ is a separating family of seminorms on $X$. Denote by $\sigma(X,P)$, the weak topology on $X$, the smallest topology on $X$ that makes each ...
0
votes
1answer
42 views
Operator Norm of a Linear Transformation
PROBLEM For the linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R}^m$ equipped with the $l^1$-norm, namely, for $x\in\mathbb{R}^n$, $||x||=\sum_{j=1}^n |x_j|$ and similarly for ...
3
votes
2answers
114 views
Can the $0$-norm represent determinism?
In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm.
Call $\{v_1,\ldots,v_N\}$ a unit vector ...
2
votes
2answers
67 views
Hahn-Banach theorem (second geometric form) exercise #2
Let $X$ be a Hilbert space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that
$$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F),$$
and any kernel of the involved functionals is ...
1
vote
2answers
41 views
What happens when you change space of test functions associated with weak derivatives?
Recall that $u \in L^2(0,T;H^1)$ has weak derivative $u' \in L^2(0,T;H^{-1})$ iff
$$\int_0^T uv' = -\int_0^T u'v$$
holds for all $v \in C_0^\infty(0,T).$
What happens if we only require that this ...
0
votes
1answer
59 views
Non-rectifiable space-filling curve
Check that the following curves γ : [0, 1] → R^2 are not rectifiable
(a) γ(0) = (0,0) and γ(x) = (x,xsin(1)) for x≠0.
x
(b) Îł is a space-filling curve: by this we mean that the image of the ...
3
votes
0answers
59 views
Computing the Fourier transform (in the sense of tempered distributions) from a sequence of samples
Let $f\in L_{loc}^2 (\mathbb{R})$ be a function of period $2\pi$. I'd like to get some help proving the following identity:
$\hat{f}=\Sigma_{n=-\infty}^{\infty} \hat{f}(n)\delta_n$
With some ...
0
votes
1answer
36 views
How to use the Mean Value Theorem to find the “Contraction Constant”
Show that the contraction $T(x)= (1+x)^{1/3} $ on the interval $I=[1,2]$ satisfies the definition of a contraction.
It's not just this problem-- on this site and others explanations will say "the ...
3
votes
1answer
181 views
Question about theorem 3.2 from Morse theory by Milnor
THe demonstration of the theorem 3.2 in the book Morse theory by Milnor
is given in the special case whene the manifold is the Torus ,
My question is : can i prove it in the case where the ...
3
votes
1answer
37 views
Let $(X,\tau)$ be compact Hausdorff with $C(X,\Bbb R)$ finite dimensional. Show that $X$ is finite
If $(X,\tau)$ is a compact Hausdorff topological space so that $C(X,\mathbb{R})$ is finite dimensional real vector space, would any one help me to show $X$ is finite set? $C(X,\mathbb{R})$ denotes the ...
2
votes
1answer
26 views
norm of product of normed spaces
If $(X_1,||.||_1)$ and $(X_2,||.||_2)$ are two normed spaces and define norm on $X_1\times X_2$ as $||x||=\max(||x_1||_1,||x_2||_2)$. I want to check the triangle inequality property for this norm, ...
0
votes
1answer
24 views
prove that linear span of an orthonormal set M of a hilbert space is closed
prove that linear span of an orthonormal set M of a Hilbert space is closed
I think i need a convergent seq in M and show that the limit belongs to span of M. but could not do it.
0
votes
2answers
31 views
Inner product convention for $\ell^p$?
So I'm reading through some analysis problems and one is discussing $\ell^p$ (the space of $p$-summable sequences $x: \mathbb Z^+ \to \mathbb C$ such that $\sum_{n \in \mathbb Z^+}|x_n|^p < ...
1
vote
2answers
61 views
Show existence of continuous functions $f$ with $f''=\delta_0-\delta_1$
Let $u$ be the distribution on $\mathbb{R}$ given by $$u=\delta_0-\delta_1 $$
(a) show there exists a continuous function $f$ such that $f''=u$ and indicate such one. I thought of doing this with ...
4
votes
1answer
76 views
Hahn-Banach theorem (second geometric form) exercise
Let $X$ be a vector normed space and $\{F,F_1,\ldots,F_N\}$ linear functionals over $X$ such that
$$\bigcap_{i=1}^N\mbox{ker}(F_i)\ \subseteq \mbox{ker}(F).$$
Apply the Hahn-Banach theorem (second ...
1
vote
1answer
37 views
PDEs: subsequence converges to solution, so whole sequence does too
Suppose we want existence of a function $u$ for the PDE
$$(\frac{d}{dt}u,v) = b(u,v)$$
for all $v$ in a test space.
Sometimes in PDE you have use a Galerkin approximation, so say $u_n$ is a sequence ...
2
votes
1answer
127 views
An other question about Theorem 3.1 from Morse theory by Milnor
In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that:
for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla ...
0
votes
0answers
49 views
Properties of the derivative $\frac{\delta F(\phi(x))}{\delta\phi(y)}$ , where $ \phi(x) = F(\phi(x)) $ is a fixed point problem.
Dear experts I have a fixed point problem of the type:
$ \phi(x) = F(\phi(x)) $, where $x \in \mathbb R^3 $.
$\phi(x)$ is differentiable non-negative function on a given domain. I am trying to find ...
1
vote
1answer
33 views
Every almost periodic function is uniformly continuous
I know that weakly almost periodic functions an a locally compact group are uniformly continuous. But I do not know how to prove it. Would you please introduce a good reference to me? Thanks.
1
vote
1answer
73 views
Need help in showing that $F(x)/x^{1/q}$ goes to $0$ as $x$ goes to $0$ and $\infty$.
$1<p<\infty$, $f\in L^{p}(0,\infty)$, $p^{-1}+q^{-1}=1$, define $$F(x)=\int_{0}^{x}f(t)dt,$$ then I need to show that $\frac{F(x)}{x^{\frac{1}{q}}}\rightarrow 0$ as $x\rightarrow 0$ and ...
1
vote
3answers
95 views
Continuous Function on a Closed Bounded Set in $\mathbb{R}^n$ then that function is bounded and uniformly continuous
Theorem : Let $A$ be closed bounded set in $\mathbb{R}^n$, and let $f:A\rightarrow\mathbb{R}$ be continuous. then $f$ is bounded and uniformly continuous on $A$.
I've been proved this theorem, my ...
3
votes
1answer
32 views
Examples of some linear and nonlinear operators
Let $H$ be a Hilbert space.
Could you please give me examples of linear or nonlinear operators $F: H \to H$ such that
$$
\limsup\limits_{\|x-y\| \to 0} \|F(x)-F(y)\| = +\infty \quad \forall x,y\in H ...
7
votes
0answers
79 views
Existence of a map in a Hilbert space
Let $H$ be an infinite-dimensional Hilbert space, $B$ be its unit ball: $B=\{x\in H: \, \|x\|\leq 1\}$.
Does there exist a continuous map $f:H\to H$ such that $f(f(x))=x$ $\forall x\in H$, $f$ has no ...
3
votes
2answers
83 views
Derive the solution to the Lagrangian $ \mathcal L= y(x)\sqrt{1+y'(x)^2}$
I am supposed to derive the solution to the Lagrangian $$ \mathcal L= y(x)\sqrt{1+y'(x)^2}$$
Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be ...
1
vote
0answers
34 views
Unbounded self- adjoint and von Neumann algebra
I am reading Conway's Functional Analysis. Here is one exercise problem.I don't know how to show the following fact. For unbounded self-adjoint $T$ in Hilbert space $H$
1) $T$ commutes with its Borel ...
1
vote
1answer
39 views
Extension of Fourier Transform
We know that Fourier transform $ \mathcal{F} : L^1 \rightarrow C_0 $ can be extended to $ \mathcal{F} : L^2 \rightarrow L^2 $ which forms a unitary isomorphism from Plancharel Theorem. Hence as for $ ...
2
votes
2answers
35 views
Properties of an integral operator: $(Au)(x)=\int_{\bf R}e^{-|x|-|y|}u(y)dy$
My questions are motivated by the following exercise:
Consider the eigenvalue problem
$$
\int_{-\infty}^{+\infty}e^{-|x|-|y|}u(y)dy=\lambda u(x), x\in{\Bbb R}.\tag{*}
$$
Show that the spectrum ...
2
votes
0answers
23 views
what to do if it's not direct sum?
Suppose $X=Y+Z$ is Banach, $Y$ and $Z$ are closed subspaces. I want to show there exists $\alpha>0$ such that $\forall x \in X, \exists$ $y \in Y$ and $z \in Z$ such that $x=y+z$ and $\|y\|+\|z\| ...
0
votes
0answers
39 views
Convex Hull of Precompact Subset is Precompact
I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact.
I've come across a ...
1
vote
1answer
45 views
Is this set $\{ p(x): x\in \operatorname{bco} A\}$ bounded in $\mathbb{R}$?
$\newcommand{\bco}{\operatorname{bco}}$Here are some terminologies.
Definition. Let $X$ be a real vector space and let $A\subseteq X$. The balanced-convex hull of $A$, denoted $\bco A$, is the ...
1
vote
1answer
36 views
The set of compact linear operators is a subspace of the set of bounded linear operators
I know that a linear operator $T:X \to Y$ (where $X$ and $Y$ are normed vector spaces) is compact if for every sequence
$\left(x_{n}\right)\subseteq X$ s.t. $\left\Vert x_{n}\right\Vert \leq C$,
the ...
