Complete normed spaces whose norm comes from an inner product.
1
vote
1answer
74 views
Riesz Representation theorem-pde
Consider $\sum_{i,j=1}^n \displaystyle\int_{\mathbb{R}^n} \dfrac{\partial^2 u}{\partial^2 x_i} \overline{\dfrac{\partial^2 v}{\partial^2 x_j} } dx + \lambda \displaystyle\int_{\mathbb{R}^n} u ...
2
votes
2answers
83 views
True or False; Functional Analysis
Given $T: V \to W$ with $V,W$ being Hilbert Spaces. We always have $\| T^ *\| = \| T \|$.
I think it is true because of Riesz' Theorem, but I am not sure if a proof is necessary.
EDIT: In case ...
6
votes
1answer
50 views
Approximating a Hilbert-Schmidt operator
Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$
where ...
2
votes
1answer
147 views
An orthonormal family in an inner product space
Why does the inner product space $( C[0,1], \| \cdot \|_2$) have an orthonormal family $(e^{\color{red}{2\pi}inx})_{n\in \mathbb{N}}$ ?
1
vote
1answer
31 views
Why is $\langle f, u \rangle_{H^{-1}, H^1} = (f,u)_{L^2}$ when $f\in L^2 \cap H^1$ and not $\langle f, u \rangle_{H^{-1}, H^1}=(f,u)_{H^1}$?
More generally, if $V \subset H \subset V'$ are Hilbert spaces, why is $$\langle f, u \rangle_{V',V} = (f,u)_{H}$$ when $f\in H \cap V$ and not $$\langle f, u \rangle_{V',V}=(f,u)_{V}?$$
Is this what ...
3
votes
0answers
64 views
Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.
For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$
Here is a quick one. If someone could improve this it would be great
Proof
By Cauchy Schwarz, $\langle x,z \rangle ...
0
votes
0answers
8 views
Weak limits and structure of a generated semigroup
I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following:
Theorem. (Jacobs–Glicksberg–de Leeuw ...
1
vote
1answer
66 views
Bounded linear operator in weak topology
Let $B$ be a bounded linear operator on $H$. Prove $B\colon (H,w)\to (H,w)$ is continuous. $(H,w)$ is a Hilbert space with its weak topology.
4
votes
1answer
138 views
Brownian Motion Covariance: max instead of min
It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion.
Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
1
vote
1answer
22 views
Is the Strong Limit of a Linear Operator in a Hilbert Space the Same as the Norm Limit?
If $H$ is a Hilbert Space, and I have an operator $F:H \rightarrow H$ which is the limit of a sequence of operators $F_n$ with respect to the operator norm; and this same sequence of operators ...
5
votes
0answers
223 views
Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)
As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
4
votes
4answers
275 views
Weak and pointwise convergence in a $L^2$ space
Let $I$ be a measured space (typically an interval of $\Bbb R$ with the Lebesgue measure), and let $(f_n)_n$ a sequence of function of $L^2(I)$.
Assume that the sequence $(f_n)$ converge pointwise ...
1
vote
1answer
40 views
Weak convergence-exercice
Let $\Omega$ be an open set in $\mathbb{R}^n$ and let $(u_n)$ be a bounded sequence in $H^1_0(\Omega).$
Who's the theorem say that we can extract a subsequence denoted $u_{n}$ as $u_n$ weakly ...
2
votes
2answers
68 views
If $Lat(\mathcal{A})$ is trivial then $\mathcal{A}'$ consists of scalars.
This is related to Exercise 3 of Section 2.5 of Arveson's book on spectral theory. For those who are interested, we were asked to show the following
$\mathcal{A}$ is a Banach *-algebra. ...
1
vote
1answer
28 views
Functional on Triangulation
Let $\Omega$ a bounded open subset of $\mathbb{R}^2$ and let $\mathcal{T} :=\{T_1,T_2,\ldots,T_N\}$ a triangulation of $\Omega$, i.e,
$\overline{T}_i$ is a triangle with non empty inner, $\forall ...
0
votes
1answer
50 views
Dual space of product of Hilbert spaces
What can we say about the dual space of the product of Hilbert spaces? Suppose $V = A \times B$ and the inner product of two vectors in this space is the obvious one (add the two separate inner ...
2
votes
1answer
141 views
Complex conjugate of the Hilbert space
Consider a Hilbert space $H=L^2(\mathbb{R}_+)$, take its conjugate $\overline{H} := \left\{f^{+}, f \in H \right\}$, where $+$ stands for the conjugation. Space $\overline{H}$ is a Hilbert space with ...
0
votes
1answer
17 views
What is bounded point evaluation property?
I read an article, it contains sentence like this - The hilbert space A possesses the bounded point evaluation property. What does this mean? I found this Meaning of Point Evaluation, is it connected ...
1
vote
1answer
43 views
Study the equivalence of these norms
I have two Hilbert spaces $H_1$ and $H_2$ and I consider a set of functions $f$ which decompose as $f=g+h$ with $g\in H_1$ and $h\in H_2$. I know that this decomposition is unique.
So I define the ...
2
votes
1answer
68 views
Matrix Representation of Trace Class Operators
Suppose we have a separable Hilbert space (thus with a countable basis) and that represent an operator in matrix form, i.e:
$A: H \rightarrow H $$$x \;\rightarrow \sum_{j \in \mathbb{N}}\left(\sum_{k ...
2
votes
1answer
102 views
Point spectrum in Hilbert spaces
Let $H$ be a Hilbert space and and $T\in B(H)$ be normal and $\sigma_p(T)$ be the point spectrum of $T$ (i.e the set of all eigenvalues of T) and let $E$ denote the spectral measure. I'm trying to ...
2
votes
1answer
205 views
Hellinger-Toeplitz theorem use principle of uniform boundedness
Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...
0
votes
0answers
26 views
Use uniform boundedness to show self adjoint operator is bounded [duplicate]
Suppose we have an operator T on a Hilbert space $H$, such that for all $x$, $y$ in $H$, $<x,T(y)>$ = $<T(x),y>$. How then can we use the uniform boundedness principle to show that T must ...
2
votes
1answer
53 views
Two inequalities related to norm
We have some difficulties in the following problem:
Let $H$ be a real Hilbert space.
Find $\alpha>0$ such that
$$
\langle\frac{u}{\sqrt{\|u\|}}-\frac{v}{\sqrt{\|v\|}}, u-v\rangle\geq
...
1
vote
0answers
30 views
How can projection operators be limits of powers of unitary operators?
Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact ...
2
votes
1answer
62 views
Show for compact operator $K$, if $||Kf|| < ||f|| \forall f$, then $||K|| < 1$.
I wanted to check my reasoning on proving this statement, and see if anyone had suggestions for other proofs of this fact.
Again, the statement is, if $K$ is a compact operator on a Hilbert space ...
1
vote
0answers
40 views
Verify solution: Is this gradient, correct?
For a function $$f(X)=\operatorname{tr}(X^TAX)+\|\operatorname{diag}(X^TX)-\alpha I\|_2,$$ where all entries are real and $\alpha$ is a real scalar, while $A$ is a p.s.d matrix and $X$ is a real ...
0
votes
0answers
18 views
Geodesic on a Hilbert manifold
Given a Hilbert manifold $\mathcal H$ (always using the natural Hilbert inner product) and a geodesic $\Gamma(t)$ in this manifold, can one show that the projection of this geodesic onto a submanifold ...
1
vote
1answer
45 views
Is this function in the space $L^1$?
I have this function
$$f(x)=\frac{1}{\vert x-y\vert^2(1+\vert x\vert^2)^s}$$
with $x\in\mathbb{R}^3$ and $y$ a fixed point. I have to study for which values of $s>0$ it belongs to ...
2
votes
0answers
62 views
Prove Heisenberg uncertainty principle (measure and integration theory)
Here is a question in measure and integration theory,
Let $f$ be a continuously differentiable complex function on $\mathbf{R}$ s.t. the functions $x \mapsto xf(x)$ and $f'$ are in ...
2
votes
1answer
24 views
Quick question about sum of subspaces of a Hilbert space
I just have a quick question. Suppose there is $Z$, a Hilbert space, with $A$ and $B$ closed linear subspaces. If $(a,b)=0$ for all $a \in A$ and $b \in B$, I know that $A+B$ is also closed. I don't ...
0
votes
2answers
51 views
Extending bilinear form from subspace to whole space
Let $X$ be a linear subspace of a Hilbert space $Y$. Let
$a(\cdot,\cdot):X \times X \to \mathbb{R}$ be bilinear. Suppose I know what $a$ is on $X$.
Is there some theorem or other that tells me that ...
0
votes
0answers
47 views
Showing an operator is self adjont
I am trying to show that the operator:
$$Tf(s)=5s^2\int_0^1t^2f(t)dt+2\int_0^1f(t)dt$$ is self adjoint where $H=L(0,1)$ with real scalars and $t\in \mathcal{L}(H)$.
So I can re-write this operator ...
1
vote
1answer
60 views
Calculating the Norm of an operator in $L^2(0,1)$
If I have the following operator for $H=L^2(0,1)$:
$$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this:
I know that in $L^2(0,1)$ we have that ...
0
votes
1answer
32 views
Intersection of affine subspaces of finite codimension in Hilbert space
I'm wondering whether the following assertion is true:
Any two affine subspaces of the same finite codimension in a ($\infty$-dimensional) Hilbert space either are parallel or have nonempty ...
7
votes
1answer
59 views
Is $\mathcal{C}([0,1])$ homeomorphic to a Hilbert space?
Let $\mathcal{C}([0,1])$ the Banach space of continuous functions from $[0,1]$ to $\mathbb{C}$. The norm on $\mathcal{C}([0,1])$ is $f \mapsto \| f\|_{\infty}= \sup_{x \in [0,1]} |f(x)|$.
Is it ...
2
votes
1answer
42 views
Is my proof correct? I want to show if $V \subset H$ is dense, then $L^2(0,T;V) \subset L^2(0,T;H)$ is dense too.
I want to show that if $V \subset H$ is a dense embedding then $L^2(0,T;V) \subset L^2(0,T;H)$ is dense too. Everything is a Hilbert space.
Let $h \in L^2(0,T;H)$. Then $h(t) \in H$ for each $t$. By ...
2
votes
1answer
90 views
Is this set dense in $H^1(\Omega)?$
Is $$V_1 = \{v \in H^1(\Omega) \;:\;f(v) = 0 \text{ on } \partial \Omega\}$$
dense in $H^1(\Omega)$ with the same norm as $H^1(\Omega)?$
Here $f$ is some linear functional so that $V_1$ is also ...
2
votes
0answers
40 views
Prove that the sequence is in $\ell^{2}$. [duplicate]
Let $(a_{n})$ be a sequence of complex numbers such that for every $(b_{n})\in \ell^{2}$the series $\sum_{1}^{\infty}a_{n}b_{n}$ converges. Prove that $(a_{n})\in \ell^{2}.$
What I've tried so far is ...
2
votes
1answer
94 views
Matrix Representation of Operators in Infinite Dimensional (Separable) Hilbert Spaces
Suppose we have a separable Hilbert space (thus with a countable basis) and that we to represent an operator in matrix form, i.e:
$$A: H \rightarrow H \\ \; \; \; \; \; \;x \;\rightarrow \sum_{j \in ...
2
votes
1answer
115 views
Dual Space as a Hilbert Space
I have this problem:
Let $(X, \langle\cdot,\cdot\rangle)$ a Hilbert Space on $\mathbb{R}$
with Riez map $\mathcal{R}:X^{\prime}\rightarrow X$, define
$[\cdot,\cdot]:X^{\prime}\times ...
2
votes
1answer
54 views
Unbounded operator $T $ is bounded below when $\overline T$ is bounded
How to prove the following?
A densely defined symmetric operator $T$ in Hilbert space $H$ has a closure $\overline T$ which is bounded iff both $T,-T$ are bounded below (there exist constants $c,c' ...
2
votes
3answers
43 views
The Kernel of unbounded operator in Hilbert space
If $T$ is a densely defined operator from a subspace of a Hilbert space $H$ to a Hilbert space $K$, how to prove that $\mbox{Ker}(T)=\mbox{Ker}(T^*T)$?
1
vote
1answer
33 views
Does a cofinite dimensional subspace of a subspace remain cofinite dimensional upon taking closures?
Let H be a separable, infinite dimensional Hilbert space. Let X and Y be (not necessarily closed) subspaces such that X is a cofinite dimensional subspace of Y. Let X′ be the closure of X and Y′ the ...
2
votes
1answer
46 views
inner product space and injective -surjective
Let $V$ and $W$ be two finite-dimensional inner product spaces over the same field
and let $T\in \mathcal{L}(V,W)\ $ be a linear transformation. Show that $T$ is injective if $T^*$ is surjective.
2
votes
2answers
82 views
Are WOT/SOT topologies hereditarily separable?
Just out of curiosity,
Are weak and strong operator topologies on $B(H)$ hereditarily separable?
In other words, if $S$ is a subset of $B(H)$, where $H$ is a separable Hilbert space, is $S$ ...
6
votes
1answer
65 views
Tight Probability on Hilbert space
I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$
And for any ...
4
votes
1answer
132 views
Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces
I am trying to prove the existence of a weak solution of the problem:
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
on the bounded open set $U\subset\mathbb{R}^n$ ...
1
vote
0answers
31 views
Closed unit ball in infinite dimensional normed linear space
I have to prove that in any infinite dimension normed linear space we have that the closed unit ball is not compact.
I know that I have to construct a sequence such that $||x_n||=1$ and ...
2
votes
0answers
46 views
Find a bounded function with a supporting point
Given, $g(Z)=\operatorname{tr}\phi(Z)$, where $\phi(Z)= Z^T\left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right) Z$ where $Z$ is a real rectangular matrix with more rows than columns (tall and ...
