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

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5
votes
1answer
580 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
202 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
71 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
62 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
140 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
137 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
254 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
137 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
89 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
139 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
58 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 ...
4
votes
1answer
376 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
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 $A=B$, ...
0
votes
2answers
144 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
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 ...
16
votes
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
votes
2answers
207 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
235 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
49 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
74 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
111 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
63 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 ...
1
vote
0answers
36 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
votes
1answer
65 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
votes
1answer
123 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
vote
1answer
52 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, ...
3
votes
2answers
70 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 $H$?
1
vote
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
votes
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
votes
0answers
63 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 ...
1
vote
2answers
140 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
votes
0answers
90 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
104 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
votes
3answers
750 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?
0
votes
1answer
183 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
vote
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
vote
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 ...
3
votes
2answers
56 views

Question about finding minimum-Hilbert spaces

How to find $$\min_{a,b,c\in\mathbb{C}}{\int_0^{\infty}} |a+bx+cx^2+x^3|^2 e^{-x} dx = ?$$ Thanks in advance.
0
votes
1answer
35 views

Finding distance in Hilbert space

How to calculate $d(e_1,L)$, where $e_1=(1,0,0,\ldots)$ and $L=\left\{x\in l^2\mid x=(\xi_j)_{j=1}^\infty,\sum_{j=1}^n\xi_j=0\right\}$. Thanks in advance.
3
votes
1answer
486 views

A strictly positive operator is invertible

Suppose that $H$ is an Hilbert space, and $T: H \to H$ is a self-adjoint strictly positive operator (i.e. $\langle Tx,x\rangle > 0$ for all $x \neq 0$). How do I show that this operator is ...
1
vote
1answer
226 views

Operator self-adjoint

I have this paragraph : "Let M be a Hilbert-Riemannian manifold. $f \in C^2(M,R), p \in K$ is called a nondegenerate critical point, if $d^2 f (p)$ has a bounded inverse. Since $A = d^2 f (p)$ is a ...
3
votes
0answers
77 views

Is this operator bounded?

Let $w_j$ be a basis ( not orthogonal) of the Hilbert space $H$. For $h = \sum^\infty a_iw_i$ define $P_n(h) = \sum_{i=1}^n a_iw_i$. Is this operator bounded in $H$ I don't think it is but I feel ...
1
vote
2answers
324 views

Projection operator for non-orthonormal basis

Let $V \subset H$ be Hilbert spaces. Let $\{v_j\}_{j=1}^\infty$ be a basis for $V$ and $H$. Define $V_N$ to be the span of $\{v_j\}_{j=1}^N$. We can define a projection operator $P:H \to V_N$ by ...
2
votes
1answer
84 views

Does a continuous embedding preserve gaps between subspaces?

I have a separable, reflexive Banach space $(V,\|\cdot\|)$ that is continuously and densely embedded in a Hilbert space $(H,|\cdot|)$. This means, there is a bounded linear injection map $j\colon V ...
3
votes
1answer
83 views

How to prove that the set $e^{inx}$ is closed with respect to this measure?

Why is the set $\{e^{inx}\}$ closed in $L^2(d\nu)$, where $d\nu(x)=(1+|h(x)|)dx+|ds(x)|$? $d\nu$ is defined in this proof I am struggling to understand, where $\mu$ is a complex Borel measure on ...
3
votes
2answers
85 views

What is the correct definition of the cuspidal subspace of $L^2$?

I have a few (semi-)related questions regarding certain Hilbert space representations of locally compact groups that come up in the theory of automorphic forms. Let $G$ be a unimodular locally ...