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, ...
1
vote
1answer
28 views
Characterization of $T+T^*\geq 0$, for $T$ a bounded operator on Hilbert space
(This is Exercise 3.2.1 in Pedersen's book Analysis Now.) Let $T$ be a bounded operator on a complex Hilbert space $H$. I want to prove that $T+T^*\geq 0$ if and only if $T+I$ is invertible in ...
-2
votes
0answers
31 views
weakly closed sets and converging sequences
Let $E$ be a normed space. $C$ is a weakly closed set in $E$ if and only if whenever $x_n\in C$ converges weakly to some $x$ we have $x\in C$.
0
votes
0answers
17 views
Analogue of continuous mapping theorem for convergence in $L^2$
Could you help please:
Is there any analogue of continuous mapping theorem for convergence of sequence of random variables in $L^2$?
I mean:
If $g$ is a continuous function (not differentiable in ...
1
vote
0answers
32 views
A trouble about the Simons’ inequality
I have a trouble in the proof to Simons’ inequality:
About prove that:
$\displaystyle \inf_{x \,\in\, C_1} \sup_{B} (x) \le \sup_{B} (\lim_{n} \sup (x_n)) \Longrightarrow \sup_{B} (\lim_{n} \sup ...
5
votes
0answers
28 views
Convergence of eigenvalues for sequence of compressions of a compact operator
Suppose $H$ is a separable Hilbert space, $A$ is a Hilbert Schmidt operator on $H$, and $P_n$ is an increasing sequence of finite rank orthogonal projections of $H$ (so $P_nx\rightarrow x$ for all ...
3
votes
0answers
51 views
Question on derivative
I have this :
And i don't understand (3.5) .
i.e : why $\displaystyle\frac{d}{dt} G_t(\eta(t)u)=(G'_t(\eta),\eta ')+\partial_tG_t(\eta))$
Please
Thank you .
0
votes
0answers
15 views
Convolution - well-defined operator [duplicate]
How to explain (to show) that the convolution (operator $\left( Tf\right)$ ) $\begin{equation*}
T:L_{p}\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) \rightarrow ...
0
votes
0answers
24 views
Properties of generalized Toepliz operators
I am going to conference where I would like to say a few words about the properties of generalized Toeplitz operators.
If someone could tell me where i can find information about this ...
1
vote
2answers
32 views
Brouwer's fixed point theorem (for unit balls) and retractions
Let $X$ be a generic Banach space over $\mathbb R$ (also infinite dimensional), and let $B$ be the unit ball in $X$.
I want to prove that the following proposition
$B$ is a fixed-point space if ...
4
votes
0answers
51 views
Isometry on a dense sub-space of a Banach space?
Let $X$ be a Banach space and let $D$ be a dense sub-space of $X$. I don't know if the following fact is true:
Fact: For every (linear) isometry $T\in Iso(X)$ and for every $\varepsilon > 0$ there ...
2
votes
1answer
34 views
A trouble about the Ekeland variational principle
I have a trouble in the proof to $EVP$ theorem:
About the existence of the $\lim (\varphi(y_n))$ ?
Any hints would be appreciated.
1
vote
1answer
62 views
Constructing a functions with Gelfand Naimark
If $X$ and $Y$ are compact Hausdorff spaces, show that for any algebra homomorphism
$$
F:C(Y) \to C(X)
$$
there exists a continuous function $f:X\to Y$ such that
$$
F(\phi)=\phi \circ f, \forall \phi ...
0
votes
1answer
37 views
Application of Stone-Weierstrass with a non-unital algebra
Let $X$ be a locally compact Hausdorff space. We say that a function $f: X \to \mathbb{R}$ vanishes at $\infty$ if for each $\epsilon >0$ there exists a compact $K_\epsilon \subset X$ such that ...
2
votes
1answer
36 views
Partial differential equation - regularity question
suppose $\Omega \subset \mathbb{R}^n$ open and bounded and $\partial\Omega\in C^{4,\gamma}$. I consider a boundary value problem in the form
$\begin{cases}
\Delta^2 u(x)=f(x)-u|u|^{p-1} ...
3
votes
1answer
38 views
Are weak* topology and strong topology the same in $L^\infty$?
Let $(\Omega, \mathcal{F}, R)$ be a reference probability space.
For short, we use $\mathbb E[\cdot]$ to denote the expectation operator
$\mathbb E^{R}[\cdot]$ under probability $R$.
We consider the ...
2
votes
0answers
47 views
Deleting “weak homeomorphism” in a Hilbert space
It is well-known that there exists a homeomorphism $h$
from an infinite-dimensional Hilbert space $H$ to $H\setminus\{0\}$.
Does there exist a "weak homeomorphism" $g:H \to H\setminus\{0\}$,
that is, ...
1
vote
1answer
22 views
Given a specefic set $ A$ we need to find $A^\perp$
Suppose we have a set of functions which are an element of $L^2[0,1]$ where if we let f(x) be the function equal to 0 from $0<x<1/2$. If this set A is a subset of the Hilbert space $L^2[0,1]$ ...
6
votes
0answers
71 views
Explicit form of the homeomorphism between $C[0,1]$ and $C[0,1]\setminus 0$
How to construct the homeomorphism between $C[0,1]$ and $C[0,1]\setminus\{\theta\}$
in the explicit form?
Here, as usual, $C[0,1]$ is the Banach space of continuous functions $f:[0,1]\to\mathbb{R}$ ...
4
votes
0answers
57 views
Is this a spectral decomposition/embedding/isometry?
Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$.
Now if I take the same ...
3
votes
2answers
64 views
Relations between $\|x+a\|$ and $\|x-a\|$ in a normed linear space.
1) Can it happen that $\|x+a\|=\|x-a\|=\|x\|+\|a\|$ when $a\ne0$?
2) How large can $\min(\|x+a\|,\|x-a\|)/\|x\|$ be when $\|x\|\ge \|a\|$?
(For a inner-product space, the answers are no and ...
1
vote
1answer
30 views
Show that the subspace A is the whole Hilbert space H
"Let $A$ be a subset in a Hilbert space $H$, such that $x\in H$ and $x \perp A$ imply $x = 0$.
(1) Show that the closed subspace that is generated by $A$ is $H$.
(2) Let $f(x)$ be a square summable ...
1
vote
1answer
36 views
How many methods for smoothing an unsmoothed function?
Which is the simplest one? For example, we smooth $f(x)=|x|$ to
$$f(x)=\begin{cases} \frac{x^2}{\epsilon}+\frac{\epsilon}{2} & |x| \le \epsilon\\
|x| & |x|\ge epsilon ...
1
vote
1answer
36 views
two definitions of a bounded set in topological vector spaces
Let
$X$ be a topological vector space.A subset $E$ is bounded if to
every neighborhood $V$ of $0$ in $X$ corresponds a number $s>0$ such
that $E\subset tV$ for every $t>s$.Would the content of ...
0
votes
0answers
30 views
Supremum of product set equal to product of supremum in lattice ordered rings
Let $E$ be a commutative, lattice ordered ring, $Y\in E$ and $\mathcal X\subseteq E$, see (http://en.wikipedia.org/wiki/Partially_ordered_ring#cite_note-Henriksen-3) for a reference. Assume that ...
0
votes
1answer
30 views
Monotone operator on $L^2(0,\infty)$
I trying to prove the following assestment
Every linear monotone operator on $L^2 (0, \infty)$ is bounded
Any ideas?
Thank you
0
votes
1answer
17 views
Relationship between adjoing matrix and inverse function
I am struggling with the following excercise:
Let A be a matrix, then we have for every subspace $U$ that:
$A^*(U ^\perp)=(A^{-1}(U))^\perp$
I do not even know where to start to solve this ...
0
votes
1answer
18 views
Verifying spectral norm
I was wondering how one could verify the relation that $||A||_2 = \sqrt{\rho(A^HA)}$ for matrices. I mean I have seen this so often, but never found a proof of it. Is there a smart way to do this ...
2
votes
0answers
32 views
Meaning of nonlinear vectorial equation
I am trying to apply some methods in a paper and I have to solve the following fixed point equation from Proposition VIII.4.3 in Asmussen (2000):
$$\mu_+ =\mu ...
2
votes
1answer
37 views
How to show that T is a projection operator
For $x ∈ [0, 2π]$ let $G(x) = π^{−1}\cos x$, and define an operator $T$ on $L^2([0, 2π])$ as follows:
$$(Tf)(x) = ∫_0^{2π}G(x − x')f(x') \,dx'. $$
Show that $T$ is a projection operator.
I guess I ...
1
vote
1answer
61 views
What is the relationship between convergence uniformly, pointwisely, weakly, in $L^{\infty}$ norm and in $L^{p} $ norm?
What is the relationship between convergence uniformly, pointwise, weakly, in $L^{\infty}$ norm and in $L^{p}$ norm?
I am quite puzzled by so many convergences, can anybody tell me what is the ...
2
votes
2answers
28 views
separability of a space
When I have to show that some space $A$ IS NOT separable, does it always work if I find uncountable subset $B\subset A$, $|B|=2^{\aleph_0}$ and set C of disjoint open balls, $C=\{L(x,r): x\in B\}$.
...
1
vote
1answer
23 views
Product and Quotient rule for Fréchet derivatives
Does anyone know whether the product/quotient rule for Fréchet derivatives still hold? For example, consider the evaluation operator:
$$\rho_x : (C[a,b],\|\cdot\|_\infty) \rightarrow ...
0
votes
1answer
22 views
If $K$ is $w$−compact and convex, $f\in X^\ast \implies f$ attains its maximum on $K$
Let $X$ be a real Banach space
If $K\subset X$ is weakly compact and convex, then for a given $f\in X^\ast$ (dual space) we can always
find
$k\in K$ such that
$$\displaystyle \sup_{x\in ...
0
votes
0answers
25 views
Haar functions are an orthonormal basis of $L^2[0,1]$ [duplicate]
The Haar functions are defined by $e_0^0(x)=1$, and for $n\geq1$ y $1\leq k\leq2^n$,
$$e_n^k(x)=\left\{\begin{array}{rcl}
2^{\frac{n}{2}} & \hspace{0.125cm} & \text{if }\frac{k-1}{2^n}\leq ...
3
votes
1answer
60 views
Equivalent definitions of uniform convexity.
I think I'm dumb, but I can't follow a simple proof from "Functional analysis and infinite dimensional geometry".
They show that two different definitions of modulus of convexity of a norm are the ...
3
votes
0answers
44 views
Discontinuous points of a point-wise limit of continuous functions.
Suppose $\{f_n\}$ is a countable set of continuous functions maps $[0,1]$ to $[0,1]$ and it has a point-wise limit $f$. Does there exist such a sequence such that $f$ takes value $0$ at $\mathbb{Q}$ ...
4
votes
1answer
57 views
How can we pick $f \in C(0,T;H)$ with $f(T) =0$ and $f(0) = h$, where $h$ is arbitrary?
Let $C(0,T;H)$ be the space of continuous functions $f:[0,T]\to H$ where $H$ is Hilbert.
For every $h \in H$, why is it possible to pick a function $f \in C(0,T;H)$ such that $f(0) = h$ and $f(T) = ...
2
votes
0answers
34 views
Strictly monotone probability measure
Let $m$ be a probability measure on $X \subseteq \mathbb{R}^n$.
Let $f: X \rightarrow \mathbb{R}$ be measurable.
Assume that there exists $\epsilon > 0$ such $m\left( \{ x \in X \mid |f'(x)| \leq ...
2
votes
0answers
128 views
A bounded sequence
I have a question please :
Let $f:[0,2\pi]\times \mathbb{R} \rightarrow \mathbb{R}$ a differential function satisfying :
$\displaystyle k^2\leq \liminf_{|x|\rightarrow \infty} \frac{f(t,x)}{x}\leq ...
5
votes
1answer
42 views
A convex function that is bounded on a neighborhood is Lipschitz
Let consider a normed vector space $V$. I want to prove that
If $f:V\to \mathbb R$ is a convex function and if for some $x_0 \in V$ the function is bounded on a neighborhood $W$ of $x_0$, then ...
1
vote
2answers
51 views
Two propositions about weak* convergence and (weak) convergence
Let $E$ be a normed space. We have the usual definitions:
1) $f, f_n \in E^*$, $n \in \mathbb{N}$, then $$f_n \xrightarrow{w^*} f :<=> \forall x \in E: f_n(x) \rightarrow f(x)$$ and in this ...
2
votes
1answer
52 views
Please check my proof (that integral = 0 implies integrand = 0 a.e. in a Bochner space setting)
Let $V$ be Hilbert and separable. Suppose $f \in L^2(0,T;V')$.
I want to show that if
$$\langle f, v \rangle=0$$
holds for all $v \in L^2(0,T;V)$, then
$$\langle f(t), w(t) \rangle_{V',V} = ...
3
votes
1answer
49 views
+50
Need explanation of passage about Lebesgue/Bochner space
From a book:
Let $V$ be Banach and $g \in L^2(0,T;V')$. For every $v \in V$, it holds that
$$\langle g(t), v \rangle_{V',V} = 0\tag{1}$$
for almost every $t \in [0,T]$.
What I don't ...
1
vote
1answer
28 views
distance between a convex set and a point
Let's look at the following famous theorem:
Let $\mathcal H$ be a Hilbert space and let $C< \mathcal H$ be a (proper) closed CONVEX set. If $x_0\in\mathcal H\setminus C$ and $\eta:=d(x_0, ...
1
vote
1answer
19 views
Under what condition, $n(>2)$ non-zero vectors of equal length forms a regular n-gon in Euclidean plane
Suppose we have a unit circle in $R^{2}$ and $f_{1}, f_{2}, \cdots ,f_{n}$ be n vectors s.t. $\|f_{i}\|=1~\forall~i=1,2,\ldots,n$.\
Also we assume that $f_{1}=(1,0)$.\
Under what condition the vectors ...
3
votes
1answer
43 views
An equality in $L^2(0,T;V')$!? Weak solution to PDE via Galerkin approximations
I have the heat equation
$$u' - \Delta u = f$$
as equality in $L^2(0,T;V')$,i.e.,
$$(u',v) + (\nabla u, \nabla v) = (f,v)$$
for all $v \in L^2(0,T;V)$, where I used the same brackets for duality ...
1
vote
0answers
28 views
$\langle f, v \rangle_{L^2(0,T;H'), L^2(0,T;H)}=0$ for all $v$ implies $f = 0$?
Suppose that for some $f \in L^2(0,T;H')$,
$$\langle f, v \rangle_{L^2(0,T;H'), L^2(0,T;H)}=0$$ for all $v \in L^2(0,T;H).$
How do I show that this implies $f = 0$? $H$ is Hilbert.
2
votes
1answer
31 views
Basis for $L^2(0,T;H)$
Given a basis $b_i$ for the separable Hilbert space $H$, what is the basis for $L^2(0,T;H)$? Could it be $\{a_jb_i : j, i \in \mathbb{N}\}$ where $a_j$ is the basis for $L^2(0,T)$?
4
votes
0answers
46 views
Differentiating an infinite series in Hilbert space
Suppose $H$ is separable Hilbert space and $w_j$ is a basis. Suppose we have $h=\sum a_j(t)w_j$ an infinite sum where the coefficients are functions of $t$. The sum makes sense in the sense that the ...
4
votes
2answers
48 views
Application of uniform boundedness principle
Let $(a_n)$ be a sequence in $\mathbb{K}$ such that for each $(x_n) \in c_0$ also $(a_nx_n) \in c_0$.
Derive from the uniform boundedness principle that $(a_n) \in l^\infty$.
I see that the idea is ...
