For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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2
votes
1answer
110 views

If $V \subset H$ compact, is $L^2(0,T;V) \subset L^2(0,T;H)$ compact too?

As the question states, if we have the compact embedding of Hilbert spaces $V \subset H$, is $L^2(0,T;V) \subset L^2(0,T;H)$ compact too? If not true in general, is it true for $V=H^1(\Omega)$ and ...
2
votes
1answer
94 views

Question about Hilbert Schmidt theory

Let $V \subset H \subset V^*$ be a Gelfand triple with all spaces being Hilbert and separable. Suppose $A:V \to V^*$ is such that $$\langle Au,u \rangle_{V^*,V} \geq C\lVert u \rVert^2_{V}$$ and $A$ ...
1
vote
1answer
118 views

Is B(H) a Hilbert space?

If H is a Hilbert space, Is B(H) under the operator norm a Hilbert space? If not, is there exists any norm on B(H) that makes it a Hilbert space?
0
votes
0answers
44 views

Hilbert space basis infinite sum

If there is a basis $h_i$ for a Hilbert space $H$, can I write any $h \in H$ like $$h = \sum_{j=1}^\infty \frac{(h, h_i)_H}{\lVert h_i \rVert} h_i$$ or something like that? Basically I want a formula ...
3
votes
2answers
310 views

Eigenfunctions of Laplacian and orthonormal basis (with different inner products)

Suppose I have $L^2(\Omega)$ which has two inner products that are both norm-equivalent. The eigenfunctions of the Laplacian $\Delta$ we know forms an orthonormal basis of $L^2(\Omega)$ -- with ...
0
votes
1answer
32 views

Confusion about domains and range (operator in $L^2$)

Suppose we have bounded linear maps $F:L^2(A) \to L^2(B)$ and $G:L^2(A) \to L^2(A)$. Let $f \in L^2(B)$ and $u \in L^2(A)$. In fact suppose $f$ is smooth. Is $fF(G(u)) = F(G(fu))$? I want to say ...
0
votes
0answers
125 views

Forming the tensor product of a `real' vector space with a 'complex' vector space.

I have a question that I am hoping someone could clarify for me. Context: Consider the algebra $A = (B,\circ)$, given by: \begin{align} B = \{ \begin{pmatrix} a & f\\ \overline{f} & ...
2
votes
1answer
112 views

Is the limit of compact operators again compact?

Let $(T_n)_{n \in \mathbb{N}} \subset \mathcal{L}(\mathcal{X}, \mathcal{Y})$ where $T_n$, $n \in \mathbb{N}$, is compact. Now, assuming that $(T_n)_{n \in \mathbb{N}}$ has a limit $T \in ...
2
votes
1answer
65 views

Why is closeness of an ideal useful?

In the GNS-construction for an $C^*$-algebra $\mathcal A$ (see this script on page 30) one starts with a state $\phi:\mathcal A\rightarrow \mathbb C$ (positive linear functional with $\|\phi\|=1$). ...
3
votes
1answer
346 views

Compact operator between Hilbert spaces: range and orthogonal complement of the kernel are separable

Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be Hilbert spaces and $T: \mathcal{H}_1 \rightarrow \mathcal{H}_2$ a compact operator. I want to show that $(\ker T)^\perp$ and $\text{ran}\ T$ are separable. ...
1
vote
2answers
189 views

Hilbert space with two inner products; separability and orthonormal basis

Let $H$ be a separable Hilbert space with inner product $(\cdot,\cdot)_H$. So it has an orthonormal basis $h_j$. (You can consider $H=L^2(\Omega)$). Suppose I know that $(\cdot,\cdot)_G$ is an inner ...
1
vote
1answer
64 views

Finite dimensional subspace of Hilbert space and basis

Let $H$ be infinite-dimensional Hilbert space with basis functions $b_i$. Let $B_n = \text{span}\{b_1, ...,b_n\}$. So $\text{dim}(N) = n$. Let $c_i$ be another basis for $H$. Is it true that ...
1
vote
2answers
81 views

Linear algebra in Hilbert space

Let $M,N$ be closed subspaces of a separable Hilbert space. How to prove rigorously the following: $\operatorname{dim} M >\operatorname{dim} N => \exists u\neq0 \in M, u\in N^\perp$ ...
2
votes
0answers
108 views

Function of a completely continuous operator

I would be most thankful if you could help me with this question. If $A$ is a completely continuous Hermitian operator on a Hilbert space $H$, for what class of functions $f$ can one define a function ...
-1
votes
1answer
40 views

Range of $Df(a)$ contained in the subspace $\{f(a)\}^{\perp}$ with $f$ differentiable

Let $A$ a open set in a Hilbert Space $H$, suppose that $f:A\to F$ is differentiable at $a\in A$ and that $||f(x)||=c$ forall $x\in A$. Show that range of $Df(a)$ is contained in the subspace ...
1
vote
2answers
181 views

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$?

How to prove $[x,p]=i$ $\implies$ $[x,p^n]=inp^{n-1}$? I can do this using $p=i\frac{d}{dx}$, but my book hasn't introduced this yet so is there another proof without using this ? These are just ...
5
votes
1answer
178 views

A baby version of the Stein-Cotlar almost-orthogonality lemma

The following is an exercise from Stein and Shakarchi's Real Analysis. Suppose $\{T_k\}$ is a collection of bounded operators on a Hilbert space $H$, each with norm at most $1$. Suppose also that ...
2
votes
1answer
432 views

Direct sum of orthogonal subspaces

I'm working on the following problem set. Let $\mathcal{H}$ be a Hilbert space and $A$ and $B$ orthogonal subspaces of $\mathcal{H}$. Prove or disprove: 1) $A \oplus B$ is closed, then $A$ and $B$ ...
3
votes
1answer
98 views

Questions about $B(H)$ and $B(H)/K(H)$ as Banach space

I am trying to investigate the relation between Uniformly Convexity and existence of Schauder Basis for a Banach space. I read in a Handbook article that $B(H)$ (the algebra of all bounded operators ...
2
votes
1answer
72 views

Hilbert space on line bundle

Suppose that $L$ is a complex line bundle on a manifold $M$ with measure $\mu$, How can we prove, $L^2(M,L,\mu)$ is Hilbert space?
2
votes
1answer
78 views

Sufficient condition for self-adjoint subset of bounded linear operators on a Hilbert space being irreducible

Let $H$ be a Hilbert space and denote as $B(H)$ the bounded linear operators on $H$. Let $M$ be a subset of $B(H)$, s.t. for $A \in M$, also $A^* \in M$. How can one show that if the commutant has ...
4
votes
2answers
96 views

(From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$

Lang $SL_2$ p. 13 :Let $\{\phi_i\}$, $\{\psi_i\}$ be orthonormal bases for $L^2(X)$ and $L^2(Y)$ respectively. Let $$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$ Then $\{\theta_{ij}\}$ is an ...
0
votes
0answers
113 views

Operator identity involving square root of an operator

I would be most thankful if you could help me prove the following identity. Let $A$ and $B$ be two completely continuous Hermitian operators on a Hilbert space $H$, such that $A$ and $B$ do not ...
7
votes
2answers
262 views

Counterexample for the stability of orthogonal projections

Let $V$ be a seperable Banach space, which is dense and continuously embedded in a Hilbert Space $H$. Let $(V_m)$ be a Galerkin scheme (See definition below) for $V$. Using the embedding we can ...
3
votes
1answer
389 views

Proof Complex positive definite => self-adjoint

I am looking for a proof of the theorem that says: A is a complex positive definite endomorphism and therefore is A self-adjoint. Does anybody of you know how to do this?
5
votes
1answer
165 views

the basis for the Sobolev space $H^1_0([0,1],\mathbb{R})$

According to the Sturm-Liouville theorem, for any continuous function $p\in\mathcal{C}^0([0,1],\mathbb{R})$, there is a Hilbert basis (normlised) $(\psi_n)_{n\geq1}$ of $L^2([0,1],\mathbb{R})$ such ...
1
vote
0answers
70 views

Are the special functions independent?

maybe the bessel functions are some complicated function of the exponential function, logarithm function... or maybe there's a relation between two or more transcendental functions. Is there a way to ...
2
votes
1answer
61 views

Is $L^2(0,T;H_n)$ compactly embedded in $L^2(0,T;H)$?

Let $H$ be a separable Hilbert space with basis $h_i.$ Let $$H_n := \text{span}\{h_1,...,h_n\}.$$ Questions: 1) Is $L^2(0,T;H_n)$ compactly embedded in $L^2(0,T;H)$? 2) Is $L^2(0,T;H_n^*)$ ...
3
votes
2answers
127 views

Equivalent norms imply equivalent inner products?

Let $H$ be Hilbert and let it have two innner products $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$. If the norms $|\cdot|_1$ and $|\cdot|_2$ are equivalent, does this ever imply: there exist constants ...
4
votes
0answers
127 views

Is this projection operator onto a subspace of a Hilbert space bounded?

(I copy and paste and edit from Is this operator bounded? Hilbert space projection, my question is almost the same) Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense and ...
1
vote
0answers
30 views

Ordering a basis of a hilbert space that has 2 indices

Suppose I am told that $a_j(t)b_i(x)$ for $i,j=1,2,...$ is a orthonormal basis for a Hilbert space $H$. I want to write an element $h= \sum_{k=1}^\infty c_kh_k$ where $h_k$ is a basis for $H$ and ...
4
votes
2answers
218 views

Is this operator bounded? Hilbert space projection

Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense in $H$. Let $b_n$ be an orthonormal basis for $H$ and an orthogonal basis for $V$. Define $$P_n:H \to ...
1
vote
1answer
121 views

$\sum c_k^2<\infty$ then $A=\{\sum_{k=1}^{\infty} a_ke_k :|a_k|\leq c_k \}$ is compact

Let $\{e_k\}_{k=1}^\infty$ be an orthonormal set in a Hilbert space $H$. If $\{c_k\}_{k=1}^\infty$ is a sequence of positive real numbers such that $\sum c_k^2<\infty$, then the set: ...
2
votes
1answer
86 views

Question about bases in Hilbert spaces and subspaces

Let $H$ be a Hilbert space. Suppose I have a basis for $H$ called $\{h_j\}$. Define $$H_n := \text{span}\{h_1,...,h_n\}.$$ Suppose now I am given an orthonormal basis for $H$ called $\{v_j\}$. My ...
3
votes
2answers
126 views

Is a Hilbert space $H$ compactly embedded in its dual?

Is a Hilbert space $H$ compactly embedded in its dual? Is it compactly embedded in itself? No idea how to think of this.
0
votes
1answer
54 views

Proof that restriction of hermitian operator to its invariant subspace is also hermitian

Proof that restriction of hermitian operator to its invariant subspace is also hermitian What would be the most elegant way to prove this?
3
votes
1answer
54 views

An element of $L^2(0,T;V_n)$.

Let $V$ be Hilbert with basis $w_j.$ Let $V_n = \text{span}(w_1, ..., w_n)$. Is it true that every element $v \in L^2(0,T;V_n)$ can be written as $$v(t) = \sum_{j=1}^n a(t)w_j?$$ I think so. But my ...
3
votes
1answer
101 views

Is $L^2(0,T;V_f) \subset L^2(0,T;V)$ closed if $V_f \subset V$?

Let $V$ be an infinite-dimensional separable Hilbert space and let $V_f$ be a subspace of $V$ that is finite dimensional. It follows that $V_f$ is closed. Is it true that $L^2(0,T;V_f)$ is closed as ...
2
votes
1answer
263 views

Continuity of scalar product

In a Hilbert space $H$ with inner product and associated norm, why would if $\|x-x_n\| \longrightarrow 0$ and $\|y-y_n\| \longrightarrow 0$ also $\langle x_n,y_n\rangle \longrightarrow\langle ...
0
votes
0answers
43 views

If $V$ is finite dimensional Hilbert space, is $L^2(0,T;V)$ also finite dimensional?

If $V$ is finite dimensional Hilbert space, is $L^2(0,T;V)$ also finite dimensional? I think so, but $L^2$ is infinite dimensional so I am not sure.
0
votes
2answers
56 views

Dual spaces and subsets

Let $X$ and $Y$ be separable Hilbert spaces with duals $X^*$ and $Y^*$. We have that $Y \subset X$. Suppose $A, B \in Y^*$ and that $Ay=By$ holds for all $y \in Y$. I think this means that $A=B$, ...
0
votes
2answers
123 views

Hilbert spaces, convergent sequence

Does anybody has any idea how to proof that a a convergent sequence in Hilbert space is bounded? Thanks for help, I need this to hopefully get to understand a proof of another theorem.
0
votes
1answer
52 views

Continuous linear function agrees with inner product

Consider a continuous linear function $\lambda: H \to \mathbb{C}$, where $H$ is a Hilbert space. I want to show that there is $v \in H$ such that $$\lambda(h) = \langle h, v \rangle$$ for all $h \in ...
15
votes
4answers
747 views

How to interpret the adjoint?

Let $V \neq \{\mathbf{0}\}$ be a inner product space, and let $f:V \to V$ be a linear transformation on $V$. I understand the definition1 of the adjoint of $f$ (denoted by $f^*$), but I can't say I ...
4
votes
2answers
168 views

about weak convergence in $L^{2}(0,T;H)$

I am trying to do an exercise and if the affirmation below is true, my exercise is done . This is the affirmation : Affirmation : Let $H$ a Hilbert space and suppose $u_k$ converges weakly to $u$ ...
5
votes
3answers
210 views

Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
2
votes
1answer
48 views

Prove that there $B : H\to H $ bounded such $ B^n = A $.

Let $ A : H\to H $ a compact self-adjoint operator. Suppose $ A $ is positive. let $ n \geq 2 $. Prove that there is $B : H\to H $ bounded such $ B^n = A $.
2
votes
0answers
72 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, ...
5
votes
0answers
107 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 ...
1
vote
1answer
57 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 ...