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
2answers
88 views
Show existence of a continuous $k$ on $\mathbb{R}^2$ such that $(u,\phi)= \int_{\mathbb{R}^2}k\phi dx$ for all $\phi$
(b). Let $u$ be a distribution on $\mathbb{R}^2$. Assume there exists a continuous function $h$ on $\mathbb{R}^2$ such that $(u,\Delta \phi) = \int_{\mathbb{R}^2}h\phi dx $ for all $\phi\in ...
0
votes
2answers
47 views
Self adjoint positive definite and product of two operators
I just wanted to ask you whether a theorem that I have found on wikipedia is correct!
I have found a theorem that says, that a matrix is positive definite if and only if it is equal to the product of ...
0
votes
1answer
15 views
The definition of 1-quotient map in functional analysis
Just a quick question, i want to know the definition of 1-quotient map. I meet it in the textbook but i don't know the definition.
0
votes
1answer
64 views
$W^0$ is a subspace of $V^*$
If $W\subset V$ is a subspace of $V$, and $W^0=\{f\in V^* | f(v)=0~\forall~v\in W\}$, then how do I show that $W^0$ is a subspace of $V^*$?
0
votes
2answers
37 views
Viewing continuous linear functionals on $l^p$ spaces in component form.
So this keeps coming up for me in problems concerning properties of some $l^p$ spaces. But it seems really useful to be able to represent a linear functional f on a given $l^p$ space as $(f_1, f_2, ...
2
votes
1answer
52 views
Showing that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$
I want to show that $H_{\text{loc}}^2(\mathbb{R}^2) \subset C^0(\mathbb{R}^2)$
$C^0$ is the space of continuous functions, and $H_{\text{loc}}^2(\mathbb{R}^2)$ the set of distributions $u\in ...
2
votes
1answer
39 views
An example of self-adjoint and positive operator
Let $\Omega\subset R^2 $ and let $H=H_0^1(\Omega)\cap H^2(\Omega)$ with inner product : $\langle u,v\rangle_H =\langle u,v\rangle_{L^2(\Omega)} + \langle\Delta u,\Delta v\rangle_{L^2(\Omega)}$.
I am ...
2
votes
0answers
52 views
(localized) L^2 norm of quasimode for Laplacian
Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq ...
2
votes
1answer
55 views
Infinite limit in all points
Do there exist a Banach space (possibly nonseparable) $X$ and a mapping $F: X\to X$ such that
$$
\lim_{x\to a} \|F(x)\| = +\infty \quad \forall a\in X\quad?
$$
0
votes
1answer
27 views
Is this Nonlinear Autonomous Banach space valued ODE a flow?
I have the following analogue of Picard's theorem for Banach space valued ode's:
Let $O$ be an open subset of a Banach space $B$ and let $F$ be a nonlinear operator satisfying the following criteria
...
0
votes
1answer
55 views
self-adjoint operator proof
Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. A number $\lambda \in C$ is called an approximate eigenvalue of T if there is a sequence ${X_n} \subset D(T)$, with ...
2
votes
1answer
42 views
spectrum of unitary operator
On $L^2(-\infty, \infty)$, T is a bounded linear operator and is defined by $$Tf(t)= \begin{cases} f(t), & \text{for } t \geq 0 \ \cr -f(t), & \text{for } t<0, \end{cases}$$
I'm able to ...
2
votes
1answer
40 views
Asymptotic behaviour of solutions to elliptic PDE
Let $u$ be a solution (in the distributional sense) of
$$
\Delta u = \delta_r
$$
on $\Omega \subset \mathbb{R}^2$ open, $r \in \Omega$.
Let $w$ be a solution of
$$
Aw = \delta_r
$$
where
$A = ...
4
votes
1answer
44 views
A problem on bounded invertible linear operator in Banach space
Let $X$ be a Banach space. Let $T : X \to X$ be a invertible
linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all
$k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
-1
votes
0answers
26 views
eigenvalues of linear operators via projections
To find eigenvalues of an operator M in a Banach space (C[0,1]), M is projected in to a finite dimensional space of polynomials. The matrix of the projected operator PM can be obtained and so the ...
2
votes
2answers
56 views
$x_n\to x$ weakly for some $x$ in $X$ with $\|x\| = 1$, then show that $\|x_n- x\|^2 \to 0$.
Let $ X$ be a Hilbert space and $(x_n)$ be a sequence in $ X$. If
$\|x_n\| \le 1$ and $x_n\to x$ weakly for some $x$ in $X$ with $\|x\| = 1$,
then show that $\|x_n- x\|^2 \to 0$.
Here we just need ...
3
votes
0answers
46 views
Sub-unital maps between C*-algebras: is there any relevant result?
"In this section, we deal with positive linear maps $\phi : A \rightarrow M$ between two unital C∗-algebra $A$ and $M$ with units denoted by $I$. In fact, we may assume that $A$ is the unital ...
5
votes
1answer
62 views
Taylor series and tempered distributions
Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid?
When we interpret ...
5
votes
0answers
125 views
Operator completly continuous
For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP
consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$
and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
2
votes
1answer
35 views
How to determine the spectrum on Banach Space [duplicate]
On Banach Space $C[0,1]$, T is a bounded linear operator and is defined by $Tf(x)=\int_0^xf(y)dy$, then how can I determine the spectrum of T?
I was hinted to first show $T^nf(X)=\frac1{(n-1)!} ...
3
votes
2answers
49 views
Equivalent bounded metric: Why should one prefer $\frac{d}{1+d}$ over $\min\{d,1\}$?
This is my first question and I hope it is not considered too argumentative.
It is often useful to change the metric on a space to an equivalent bounded metric.
Traditionally, people use
$$
...
2
votes
1answer
44 views
Spectrum proofs
Let $T$ be a densely defined closed unbounded operator on a Hilbert space $H$. Show that if $\lambda$ is a point in the residual spectrum of $T$, then $\bar{\lambda}$ is in the point spectrum of the ...
0
votes
0answers
27 views
Normal endomorphism
I have a question about normal endomorphism. In class, we said that normal endomorphisms in finite dimensional real vector spaces are always of the form that we have some eigenvalues and further ...
0
votes
1answer
62 views
proof related to Hilbert Spaces
Let $T$ be a bounded linear compact operator on a Hilbert space $H$ over $C$, $A$ is a positive self-adjoint operator on $H$. How to show that $T=UA$ where $U^{+}U=I$ on the range $R(A)$ of $A$
4
votes
0answers
68 views
Solution to $\Delta_g u = \delta-1$ on a 2-sphere.
Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
0
votes
0answers
62 views
Show there exists a sequence of postive real numbers s.t. [duplicate]
Let $(f_n)_{n\in\mathbb N}$ be a sequence of measurable functions on $[0,1]$ with $\forall n\in\mathbb N, \lvert f_n(x) \rvert < +\infty$ a.e. Show there exists a sequence $(c_n)_{n\in\mathbb N}$ ...
2
votes
3answers
32 views
Uniform Convergence and differentiable functions
I have been working on this textbook question and am not sure what to do. Is there a sequence of differentiable functions on some interval, say [0,2], converging to 0 uniformly, but where $f'_n(1)$ ...
0
votes
0answers
33 views
Complete normed vector space
I came across this question and was not sure how to proceed. Is the normed vector space of all polynomial functions on [0,1] complete with respect to the infinity norm? Thank you
1
vote
2answers
53 views
Pointwise Convergence, L^2 Convergence
I was wondering if pointwise convergence of a sequence of functions implies convergence with respect to the $L^2$ norm. So if $g_n$ is a sequence of functions in $L^2[0,1]$ converging to g pointwise, ...
1
vote
1answer
19 views
Spectral family property: $\lambda \geq M \Rightarrow E_{\lambda} = I$
I'm following Kreyszig's "Introductory Functional Analysis with Applications" and I'm trying to follow his proof about some properties of a spectral family associated with a bounded self-adjoint ...
1
vote
0answers
29 views
The tightest bound on an integral
Consider a polynomial $p(x)$ where $p(x)>0$ for $x\in(0,1)$ and $p(0)=0$. Let $s(x)$ be an increasing analytic function such that $s(0)=0$ and $s(1)=1$. I am interested to bound the following ...
-1
votes
0answers
49 views
How we can change a strongly continuous semigroup to a contraction semigroup?
If $T(t), t>0$ is a strongly continuous, bounded semigroup on a Banach space. how we can transform it to contraction semigroup?
2
votes
1answer
30 views
A bound on an integral
Consider $f(z)$ and $g(z)$ where $f(z)$ is a polynomial such that $f(z)=0$
and $g(z)$ is an analytic function. I want to find the tightest bound the following integral:
$\int_0^1 f(z)g(z) dz$
I know ...
2
votes
1answer
55 views
Limit of a sequence in the space $\ell_2$
I have difficulties in the following problem.
Let $H=\ell_2$ be the space of square-summable sequences. Let $\alpha\in (0,1)$ and $\{u^k\}\subset H$ be such that
$$
u^{k+1}=(1-\alpha)u^k+\alpha ...
1
vote
1answer
23 views
Multiplicative functionals on Banach algebra closed in weak-* topology
Let $\mathcal A$ be commutative unital Banach algebra denote by $M(A)$ the space of all non-zero multiplicative functionals on $\mathcal A$.
I want to show that $M(A)$ is closed in the weak-* ...
0
votes
2answers
49 views
Self adjoint operator
I am looking in the space of test functions $ \{f \in C^\infty|f^{(n)}(a)=f^{(n)}(b)=0\};n \in \mathbb{N}_0\} $whether the n-th derivative is a self adjoint operator. the dot product is given by ...
-1
votes
1answer
25 views
An inequality : $ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$
Let $u=u(x)$ be a real-valued function defined on $\mathbb R$.
How does this inequality hold?
$$ \| u^3 \|_{H^3} \leq C ( \| u \|_{L^\infty}^2 + \| \partial_x u \|_{L^\infty}^2 ) \| u \|_{H^3}$$
...
1
vote
0answers
44 views
Direct (Inductive) limit of (locally convex) TVEs and universal property
This is not really a question, I'd just like to discuss a little about universal properties (more specifically, the direct limit) in TVEs.
I'm trying to work with universal properties in Topological ...
3
votes
1answer
39 views
Calculating average over a function set
Non-math version of the problem: I am running with a GPS device, recording my path. I know the curve the GPS has recorded. However, the GPS device actually has an accuracy, which I can assume to be ...
1
vote
0answers
32 views
adjoint operator in Sobolev space
Let $g\in H_0^1(\Omega)\cap W^{2,\infty}(\Omega),$ and let us define the operator $B : y \to g y$ from $H:=H_0^1(\Omega)\cap H^2(\Omega)$ to $H$, which we endowed with inner product : $<u,v>_H ...
5
votes
2answers
54 views
A basic question on dual space of $L^p[0,1]$
I recently started reading functional analysis on my own and have come about dual spaces and cannot get an intuitive understanding about them. This is where my intuition breaks down while ...
1
vote
1answer
52 views
exercise: limit orthonormal sequence, “Banach Space Theory”
I have an exercise from "Banach Space Theory":
Suppose $\{x^k\}_{k=1}^\infty$ is an orthonormal sequence in $l_2$, where $x^k:=(x_i^k)$. Show that $\lim_{k\rightarrow \infty} x_i^k =0 \ \forall_{i\in ...
3
votes
1answer
34 views
Is a continuous function in two variables necessarily equicontinuous in one variable?
Suppose $K \in \mathcal{C}\left(\left[0, 1\right]\times\left[0, 1\right]\right)$. Then, is it necessarily the case that the set of functions $\left\{g_y(x):g_y(x) = K(x,y), \forall y \in ...
1
vote
1answer
33 views
Gelfand transform and spectrum
Let $\mathcal A$ commutative, unital Banach algebra and denote by $\mathcal M(\mathcal A)$ the space of multiplicative functionals on $\mathcal A$. The Gelfand transform is defined by
$$\Gamma: ...
1
vote
0answers
72 views
Problem # 25, page 95, from Stein and Rami [duplicate]
Let $(X,M,\mu)$ be a measure space with $\mu(X) < 1$. Show that for any $1\le p<q$, we have $$L^q (X,\mu)\subset L^p(X,\mu).$$ Let $\ell^p(Z)$ denote the $L^p$ space of the integers equipped ...
1
vote
1answer
24 views
Extention of functions in Sobolev spaces
Let $\omega$ a subset of a domain $\Omega\in R^n,$ and let $f\in H^2(\omega)\cap H_0^1(\omega)$.
It is known that a function $u\in H_0^1(\omega)$ admits an axtention $U\in H_0^1(\Omega)$.
Does ...
0
votes
3answers
34 views
preimage definition of continuity
I'm currently studying functional analysis and the professor covered continuity using the definition that the preimage of every open set is open. I can follow the definition, which basically means ...
1
vote
0answers
34 views
Continuous spectrum for a specific linear operator
The operator is for $A:L^2[-1,1]\to L^2[-1,1]$ defined via
$$Au(x)=xu(x)+\theta\int_{-1} ^1u(s)ds.$$
The question is actually find the spectrum, but I managed to find everything else, pending the ...
1
vote
1answer
57 views
Showing that the inverse of the perturbation of a compact operator by a bounded operator remains compact.
The title says it all. If we have a Hilbert space $H$, then if $B\in \mathcal B(H)$, $L$ is a linear operator that is not necessarily bounded, $L^{-1}$ is compact, and $0\in \rho(L)\cap\rho(L+B)$, ...
1
vote
1answer
53 views
the first eigenfunction of Dirichlet problem
Let $\Omega$ be a bounded planar domain which has a axis of symmetry and $T:\Bbb{R}^2\longrightarrow\Bbb{R}^2$ symmetry with respect to this axis. Let $u_{1}(x)$ be the first eigenfunction of ...
