Tagged Questions
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, ...
0
votes
1answer
5 views
every compact subset of a TVS is bounded
I'm self-studying Functional Analysis. The following is Exercise 4.2.4 of Conway's Functional Analysis.
Let $X$ be a topological vector space. Show that every compact subset of $X$ is bounded.
For ...
51
votes
4answers
8k views
Norms Induced by Inner Products and the Parallelogram Law
Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$.
It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...
11
votes
1answer
326 views
Can a dynamic programming problem be transformed into a linear algebra problem?
Here is a simple standard economic problem:
Let Robin Crusoe have an endowment $w_0$ of lembas bread. She is immortal and discounts at a rate of $\beta$ per period. Each period (from $t=0$ on) she ...
5
votes
1answer
53 views
Can differential forms be generalized to (separable) Banach spaces?
This thought occurred to me earlier and I'm surprised I hadn't considered it previously. I get the feeling that no meaningful generalization can occur in a non-separable Banach space but on the ...
13
votes
1answer
196 views
Legendre Transformation of a Lagrangian in Classical Mechanics
I have some questions about the Legendre Transformation of a Lagrangian in Classical Mechanics to the Hamiltonian:
We start with a Lagrangian $L(q,\dot{q})=\frac{\langle \dot{q} , \dot{q}\rangle }{2} ...
1
vote
1answer
21 views
A uniform bound on $u_n$ in $L^\infty(0,T;L^\infty(\Omega))$
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. I have a sequence $u_n$ satisfying
$$\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$$
for all $n$.
Can I get a weak-* convergent ...
2
votes
1answer
46 views
Sobolev Spaces and Derivative
I need help on the problem 8.9 at page 238 of the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis.
Set $I=(0,1)$.
Let $u \in W^{2,p}(I)$ with ...
1
vote
0answers
53 views
Approximating continuous function by the span of $\{\sin(nx)\}$
Let $f $ be continuous function on $[0,2\pi ]$ and $\int_0^{2\pi} f(x) \sin(nx) \, dx =0$ for all $n$ then prove that $f$ is identically zero.
Some of my friends claim that it is not true just ...
0
votes
0answers
23 views
How to use the trace theorem to prove that a set is not empty?
Particularly, I would like an explanation of the following application of the trace theorem.
Thanks.
0
votes
0answers
18 views
Composing $L^2$-norm continuous operator and sup-norm continuous functional
Let $(X,\Sigma_X)$ be a measurable space such that $\Sigma_X$ is countably generated.
Let $B_b(X)$ be the Banach space of all bounded $\Sigma_X$-measurable functions $X\to\mathbb{R}$ equipped with ...
1
vote
1answer
92 views
Can I easily deduce this stronger spectral theorem from this weaker one?
I've just read a nice proof that:
For $T$ a self-adjoint bounded linear operator on a Hilbert space $E$, there exists a unique $C^*$-algebra isomorphism $C(\sigma (T)) \rightarrow A_T$, from the ...
0
votes
1answer
56 views
Mathieu differential equation
Given the operator $T (\psi)(x):= \psi''(x)-2q \cos(2x)\psi(x)$ with $T : D(T) \subset L^2[0,2\pi] $
I was wondering: What is the right domain $D(T)$ for this operator if we want to solve the ...
0
votes
0answers
13 views
Application Closed Graph Theorem to Cauchy problem
Consider $E:=C^0([a,b])\times\mathbb{R}^n$ and $F:=C^n([a,b])$ equipped with the product norms.
Consider
$$ u^{(n)}+\sum_{i=0}^{n-1}a_i(t)u^{(i)}=f
$$
with $$u(t_0)=w_1,\dots,u^{(n-1)}(t_0)=w_n \\
...
2
votes
1answer
53 views
Solving $ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$
Let $\phi \in \mathcal ( [0,1]^2)$ symetric , can we find a solution to the following minimisation problem?
$$ \inf \left\{ F[\nu] : \nu \in L^2 , \nu \geq 0, \int _0 ^1 \nu=1\right\}$$
with $$ ...
1
vote
1answer
36 views
Kadison's Inequality
Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional.
Is there a simple proof for Kadison's inequality:
$$\overline{\omega(A)}\omega(A)\leq\omega(A^*A)$$
1
vote
1answer
22 views
The Nash inequality on a compact manifold without a boundary.
Let $M$ be a compact manifold without boundary. Does the Nash inequality
$$\lVert u \rVert^{1+\frac 2n}_{L^2} \leq C\lVert u \rVert^{\frac 2n}_{L^1} \lVert \nabla u \rVert_{L^2}$$
or something ...
0
votes
1answer
31 views
Subtracting terms from a Fourier series
It is known that $\sum_{n=1}^{\infty}\frac{\sin(nx)}{n}=\frac{\pi-x}{2}$ in $]0,\pi]$, mostly because this is a way of evaluating $\zeta(2)$. Knowing this, is there a way to evaluate ...
1
vote
1answer
30 views
unbounded self-adjoint operator
Given an operator $T:D_1(T)\subset L^2 \rightarrow L^2$ and the same operator $T:D_2(T) \subset L^2 \rightarrow L^2$, such that the operator is both times self-adjoint and closed, with $D_1(T) \subset ...
1
vote
0answers
25 views
Sobolev spaces and using monotone convergence theorem (don't understand a paper)
I'm reading this paper. In it there the following argument (see page 240).
Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
0
votes
1answer
25 views
If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable.
If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable.
It seems very obvious intuitive, but how to write a good solid proof? Notice I take the closure of the span (the ...
-6
votes
0answers
156 views
Limit of $\arg\min$s
Let $X \subset \mathbb{R}^m$ be a compact set. For $N = 1, 2, \ldots$, let
$\, f_N : X^{N} \rightarrow \mathbb{R}_{\geq 0}$ be continuous and such that
$$ \text{for all } \{ x_n \}_{n=1}^{\infty} \in ...
2
votes
1answer
53 views
Why the space of probability measures is a subset of the measure space
Consider $\mathcal M (X)$ the measure space of a metric, compact space $X$ allowed of the weak-* topology induced by the semi-norms $\mu \in \mathcal M (X) \mapsto |\int_X f ~d\mu| \in \mathbb R ...
7
votes
2answers
386 views
Banach space valued integration (Riemann type)
Preface
The core of any notion of integral is some sort of weighted sum:
$$\sum b\mu(A)$$
Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones:
...
1
vote
1answer
44 views
Norm on unitisation of a $C^\ast$ algebra
In the theory of $C^\ast$ algebras there exists the following theorem:
If $A$ is a $C^\ast$ algebra and $\widetilde{A}$ denotes its unitisation then there exists exactly one norm that extends the ...
7
votes
1answer
116 views
Dense subspace of $L^{2}[0,1]$
I know that $C[0,1]$ is dense in $L^{2}[0,1]$ but is $\{f\in C^{2}[0,1]:f(0)=f(1)=0\}$ dense in $L^{2}[0,1]$?
4
votes
0answers
83 views
+50
Connections and dependences between topological and algebraic basis in topological vector space
On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$).
I am not sure if ...
3
votes
0answers
58 views
Positive Operators: Definition?
Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
2
votes
1answer
31 views
Strict convexity and uniqueness of functionals
Is it true that if $x$ is a norm-one vector in a strictly convex Banach space then there exists a unique bounded linear functional $f$ on that space such that $f(x)=1=\|f\|$?
It seems unlikely to me ...
1
vote
1answer
49 views
Spectrum: Polynomials
It is written in Bratteli-Robinson that some simple transformations yield the relations:
$$\sigma(a+A)=a+\sigma(A)$$
$$\sigma(A^n)\subseteq\sigma(A)^n$$
The latter one is deduced by the ...
0
votes
1answer
65 views
Momentum Operator: Selfadjoint Extensions
This might be a possible duplicate - please let me know if there is already a proof in another thread.
Consider the momentum operator on $\mathcal{L}^2[0,2\pi]$:
...
4
votes
1answer
131 views
Gilbarg Trudinger: Hölder continuity in chapter 8
I'm trying to track the behaviour of the coefficients in Theorems 8.22 and Theorem 8.24. Particularly, I'm considering the behaviour w.r.t. to the distance from $\Omega'$ to $\partial \Omega$
I'll ...
1
vote
1answer
28 views
The spectral projection of a positive operator
Let $T_{n}\in B(H)$ be a positive operator on Hilbert space $H$ and $T_{n}\rightarrow 1_{H}$ in the strong operator topology. Now fix $\delta>0$ and let $P_{n}$ be the spectral projection of ...
0
votes
0answers
14 views
Existence of Density in Bochner's Thoerem
Bochner theorem for locally compact abelian group, $G$ and a positive definite function $f$ there exist a unique measure $\mu_f$ such that:
$$f(x)=\int\limits_{\hat G}(x,\gamma)d\mu_f(\gamma)$$
Where ...
11
votes
1answer
1k views
Direct aproach to the Closed Graph Theorem
In the context of Banach spaces, the Closed Graph Theorem
and the Open Mapping Theorem are equivalent.
It seems that usually one proves the Open Mapping Theorem using the
Baire Category Theorem, and ...
4
votes
0answers
38 views
a question about Tsirelson's space
Background. Let $T$ denote the Figiel-Johnson construction of the Tsirelson space, that is, the completion of $c_{00}$ under the implicitly-defined norm
...
0
votes
1answer
26 views
Minkowski inequality for $l_p$ norm.
I'm trying to prove the Minkowski inequality for the $l_p$ norm:
$$
\| f + g\|_p \le \|f\|_p + \|g\|_p
$$
where $f,g : \mathbb{R}^n \rightarrow \mathbb{R}$ are Lebesgue measurable functions and $p ...
4
votes
1answer
86 views
Unique fixed point of a contraction defined on a closed ball which maps the boundary back into the ball
Let $X$ be a Banach space, $r > 0$, $A: K_r(X) \rightarrow X$ a contraction (where $K_r(X)$ is the closed ball of radius $r$ and center $0$ in $X$), with contraction constant $0<q<1$, which ...
0
votes
0answers
21 views
Unbounded Operators: Notation?
For continuous a.k.a bounded operators we have $\mathcal{B}(X,Y)$ stressing on boundedness and $\mathcal{L}(X,Y)$ stressing on linearity entailing $\mathcal{C}(X,Y)$.
Is there a notation for ...
1
vote
1answer
86 views
Differential equations, integral equations
Is there an analytical way of proving that if $\phi$ is a solution to
\begin{equation} y(t)=e^{it}+a\int_{t}^{\infty}\sin (t-s)y(s)s^{-2}ds,
\end{equation}
then $\phi$ would be a solution to the ...
0
votes
0answers
26 views
Two questions on tensor product
Here is a quotation of a book:
Let $M\subset B(H)$ be a von Neumann algebra and let $\pi: M\rightarrow B(K)$ be any normal representation. Then, any normal representation of $M$ can be identified ...
0
votes
0answers
23 views
Need help with the proof of the KKM-lemma.
I have been working on the proof of the KKM-lemma, which states
Let $\lbrace A_0,A_1,...,A_n \rbrace$ be a closed covering of an $n$-simplex $\sigma=[x_0,...,x_n]$ such that for each face ...
11
votes
1answer
436 views
+50
Osgood condition
Let $h$ and $g$ be continuous, non-decreasing and concave functions in the interval $[0,\infty)$ with $h(0)=g(0)=0$ and $h(x)>0$ and $g(x)>0$ for $x>0$ such that both satisfy the Osgood ...
1
vote
1answer
247 views
Prove a non-empty subset is closed in an inner product space
I hope someone would be able to help me with the finer details of this proof.
Problem:
M is a non-empty set in an Inner Product Space (IPS) X.
I need to show that the annihilator of M which is ...
1
vote
1answer
31 views
Non-existence of a continous-norm on a sequence space.
For $U\cong \prod_{n\in \mathbb{N}} \mathbb{R}$ equipped with the product topology, i have already shown, that $U$ is a Frechet-Space w.r.t. the frechet-metric. How to prove that there exists no ...
1
vote
3answers
87 views
A topological vector space with countable local base is metrizable
I feel confused by the proof of the following theorem in Rudin 2/e:
Theorem 1.24 If $X$ is a topological vector space (t.v.s.) with a countable local base, then there is a metric $d$ on $X$ s.t.
...
0
votes
1answer
47 views
Does Every Continuous Function Have an Antiderivative?
I was thinking about integrals over continuous function and wanted to know if every continuous function has an antiderivative. Generally, the only way I can think of proving this is through a linear ...
1
vote
1answer
37 views
Basis to Hilbert spaces
Let H be a separable hilbert space with orthonormal basis $\{e_i\}$. How can we construct another orthonormal basis $\{f_k\}$ such that for any k, inner product of $f_k$ with infinitely many(all?) ...
2
votes
1answer
29 views
Approximations of compact operators
Let $(\xi_n)_{n=1}^\infty$ be a sequence in a Hilbert space $K$ convergent to some $\xi$. Suppose we have a compact operator $T$ on $K$ such that $T\xi = 0$. Can we find a sequence of compact ...
0
votes
1answer
26 views
Positive-definite function on a group function on a group
I have quite a hard time understanding the definition of positive-definite functions that is based on Hilbert spaces, the one that I read from Wiki; it does not exactly specify that how $H$ relates to ...
2
votes
1answer
46 views
Composition of Partial Isometries
Let $H$ be a complex Hilbert space and $S,T \in B(H)$ partial isometries. Then $S T$ is a partial isometrie, if and only if $T^*(\ker(S)) \subseteq \ker(ST)$.
Edit:
My attempts so far:
...
