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

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2answers
210 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
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1answer
220 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 ...
4
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1answer
686 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
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1answer
110 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
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1answer
81 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
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1answer
84 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
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2answers
101 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 ...
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0answers
130 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
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2answers
284 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 ...
5
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1answer
739 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
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1answer
224 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 ...
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0answers
72 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
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1answer
66 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
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2answers
158 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
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0answers
144 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 ...
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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
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2answers
290 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
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1answer
142 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
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1answer
90 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
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2answers
152 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.
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1answer
59 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
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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
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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 ...
4
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1answer
439 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 ...
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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
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2answers
58 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 $A=B$, ...
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2answers
159 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
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1answer
54 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 ...
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4answers
1k 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
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2answers
220 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 $u$ ...
6
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3answers
246 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
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1answer
50 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 $.
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0answers
75 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
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0answers
116 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 ...
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1answer
68 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 ...
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0answers
39 views

Exercise over Haar functions

$\newcommand{\span}{\operatorname{span}}$ Define $e_{0,0}\equiv 1$, and for all $n\in \mathbb{N}$ $$e_{n,k}=\begin{cases} 2^{n/2} &\text{if } \frac{k-1}{2^n}\leq x\lt \frac{k-\frac{1}{2}}{2^n}\\ ...
4
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1answer
66 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
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1answer
127 views

if $E^2=E$, then $\text{Im}\;E\subset\left(\ker E+(\ker E)^\perp\right)$?

Notation: $V$ is a infinite-dimensional inner product space; $\langle\cdot,\cdot\rangle$ is the inner product of $V$; $E:V\rightarrow V$ is a linear map; $\text{Im}=\{E(v):v\in V\}$; $\ker E=\{z\in ...
1
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1answer
57 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, ...
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2answers
72 views

Closed linear subset of a Hilbert space

If $H$ is a Hilbert space, and if $$(a,b)_H=0$$ for every $b \in B \subset H$, where $B$ is a closed linear subset of $H$, does it follow that $a=0$, the zero element of $H$?
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0answers
34 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
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1answer
40 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
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0answers
64 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 ...
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2answers
154 views

Characterisation of norm convergence

Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$): We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and we have $x_n ...
2
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0answers
94 views

On differential geometry in Hilbert spaces

Suppose that $H$ is a Hilbert space and $M\subset H$ is a closed subset with non-empty interior and smooth boundary, whatever smooth boundary could mean. I wonder if the normal vector is onto on the ...
3
votes
1answer
110 views

The definition of addition on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ respectively. Construct the tensor product of $H_1$ and ...
4
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3answers
866 views

What is a Hilbert space?

I've just seen a question about Hilbert Subspaces. This made me wonder what a Hilbert space is. Can anyone explain in layman's terms?
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1answer
193 views

Hilbert subspace [duplicate]

Let be $H$ Hilbert space and $M\subset H$. $M=M^{\perp\perp}$ if and only if $M$ subspace of $H$. Does anyone know to prove this?
1
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1answer
56 views

$\bigcup_{n}V_n$ is dense in $V$ implies $\bigcup_{n}L^2(0,T;V_n)$ is dense in $L^2(0,T;V)$?

Let $V$ be a separable Hilbert space with basis $w_j$ and let $V_n$ denote the linear span of $w_j$ for $j=1,...,n$. Clearly $V_n$ are Hilbert spaces and $V_n \subset V_{n+1}$ for all $n$. We have ...
1
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1answer
39 views

$\bigcup_{n}V_n$ is dense in $V$ (Hilbert spaces)

I read: $\bigcup_{n}V_n$ is dense in $V$ (Hilbert spaces) Does this mean: for every $v \in V$, there is a sequence $\{v_n\}$ with $v_n \in V_n$ for each $n$ such that $|v_n - v|_V \to 0$? I ...