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, ...
2
votes
1answer
19 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
16 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
0answers
19 views
Variational formulation-exercice
let the problem $$-u'' + a(x) u = f , x \in \Omega = ]0,1[, u'(0) = u(0); u(1) = -1$$
where $f \in L^2(\Omega) , a(x) \geq a_0 > 0, a \in L^{\infty}(\Omega)$
1- Prove that the variational ...
0
votes
2answers
23 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
34 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
40 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 ...
0
votes
0answers
19 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
42 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 ...
-1
votes
0answers
26 views
Stone’s Theorem on one-parameter unitary groups
I ask this question on Mathoverflow,but they suggest me to ask it here:
What does Stone's Theorem(Stone's theorem on one-parameter unitary groups) tell us when applied to the unitary group on ...
0
votes
0answers
32 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
30 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
64 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
2answers
47 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 ...
2
votes
1answer
19 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
67 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 ...
