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

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3
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1answer
102 views

What is the image of operator exponential?

Given a Banach space $V$ and a bounded linear operator $A:V\to V$, the operator $e^A$ is bounded and invertible. When $V$ is finite dimensional, every invertible operator is of the form $e^B$ (one can ...
1
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0answers
44 views

Show that $H=\oplus_{\alpha \in \sigma(T)}\text{ker}(\alpha I -T)$

Suppose $T$ is an operator on a Hilbert space $H$ such that $\sigma(T)=\sigma_{p}(T)$ (point spectrum of $T$), and for each $\alpha \in \sigma(T)$, the corresponding eigenspace ker$(\alpha I-T)$ is a ...
1
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1answer
65 views

Product Hilbert Spaces that require completion?

My lecturer said that to make the Hilbert space $(H\oplus H',⟨\cdot,\cdot⟩ )$ we need to (1) make the cartesian product $H \oplus H' = H\times H'$, (2) give it an inner product, and - the confusing ...
0
votes
1answer
87 views

Norms for which every subset of closed unit ball containing the open unit ball is convex

It can be shown without much difficulty that any Euclidean norms satisfies the following condition :$$(P) \quad B \subset X \subset B' \Rightarrow X \, \text{is convex}$$ where $B=\{x \in E / \|x\| ...
0
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1answer
55 views

Adjoint of differential operator in two variables

I would like to find the adjoint of the operator $$L = x \frac{\partial^{2}}{\partial y^{2}} \frac{\partial }{\partial x}.$$ I know the adjoint is the operator $L^{*}$ such that $$(Lu,v) = (u, ...
0
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0answers
97 views

Comparison of Hilbert space tensor product and wedge product

For Hilbert Spaces: $$(|0\rangle + |1\rangle)\otimes (|0\rangle + |1\rangle) = |00\rangle + |01\rangle + |10\rangle + |11\rangle.$$ where all results are column vectors \begin{eqnarray*} 0 ...
0
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1answer
44 views

Calculating the form domain of an operator

I am reading the book "Mathematical Methods in Quantum Mechanics" by Gerald Teschl and just came across the concept of a form domain. It is defined for non-negative operators i.e $<\phi,A \phi> ...
1
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1answer
73 views

Identity involving the resolvent of an operator.

$\mathcal{H}$ is a complex separable Hilbert space, $D \subseteq \mathcal{H}$ is a dense subspace. $L : D \rightarrow \mathcal{H}$ is a densely defined, symmetric and closed operator. $L$ is not ...
0
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2answers
130 views

Is a Hilbert space a vector space or a space of functions?

I was learning what Hilbert space was and this is the definition that I have: $\mathcal{H}$ Hilbert Space is a vector space with $\langle \cdot , \cdot\rangle$ inner product and is complete with ...
2
votes
1answer
138 views

Uniformly bounded sequence in Hilbert-Sobolev space

Let $\Omega \subset \mathbb{R}$ be a bounded open set with $C%1$ boundary and $H^1(\Omega) = W^{1,2}(\Omega)$ be the Hilbert-Sobolev space. Let ${u_n}$ be a sequence of functions which are uniformly ...
0
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1answer
22 views

$\|x\|=1,\|x-y\|\ge\epsilon\forall x,y\in A,x\neq y$ , is this finite set in hilbert spaace?

$A$ be a subset of a Hilbert Space $H\ni \|x\|=1\forall x\in A$ and there is an $\epsilon\ni\|x-y\|\ge\epsilon\forall x,y\in A,x\neq y$ I need to know whether $A$ is finite or not. Intuitively, It ...
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0answers
25 views

How do you show the connection of reproducing kernels to feature maps?

This question is in the context of Hilbert Reproducing Hilbert Spaces and reproducing Kernels and their relation to feature maps (and machine learning). We have a Hilbert space $\mathcal{F}$ and ...
1
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1answer
29 views

Help proof regarding sequence in subset of Hilbert space

I'm to prove the following: Let $H$ be a Hilbert space, and let $M$ be a non-empty convex subset of $H$. Suppose that $(x_n)$ is a sequence in $M$ such that $ ||x_n|| \to d$, where $d= \inf ...
1
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1answer
115 views

The definition of convex body and the Hilbert cube

I currently have a question about the definition of convex body. The formal definition is: a convex body is a convex set which has non-empty interior. By non-empty interior, we meant for a set ...
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0answers
59 views

Proving that the set is closed.

We use the sequential definition to prove a set is closed. So no continuity or closure or anything related to the topology of the set is allowed. Show $A = \{ x \in \ell^2: |x_n| \leq 1/n \}$ is ...
0
votes
1answer
385 views

Existence of adjoint operator on a Hilbert space

friends! I read that the algebra $\mathscr{L}(H,H)$ of the bounded operators on a Hilbert space $H$ is a $B^\ast$-algebra in the sense defined here . I easily verify all the properties except for the ...
0
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0answers
67 views

Existance of Unitary Operator which Extends any inner product preserving operator in a Hilbert Space

Suppose $V$ is a finite dimensional Hilbert Space with a subspace $W$ Suppose $T:W\to V$ be a linear operator which preserves inner product i.e $\forall w_1,w_2\in W$ we have $\langle ...
1
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1answer
42 views

Questions about open balls and convergence in Hilbert space

So, I've started reading about dimension theory and currently am dealing with a lemma which is used in a proof of $\dim(\mathcal{\ell}^{2}_{\mathbb{Q}})=1$. This lemma says that convergence of a ...
1
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0answers
40 views

product of one dimensional basis functions spanning two dimensional space

Lets assume I have a set of basis functions $h_1(x),h_2(x), ...$ spanning the whole hilbert space of one dimensional square integrable functions. Now I want a basis set that spans the the whole ...
2
votes
3answers
140 views

Is the (first order theory) of Hilbert spaces categorical?

Suppose the axioms of a Hilbert space (i.e. vector space, inner product, completeness and separability) are formulated as a first order theory. It can be shown that any infinite dimensional Hilbert ...
0
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3answers
50 views

When a symmetric densely defined operator is an adjoint operator?

I am wondering if it is possible to say that if a symmetric differential operator is densely defined then the operator is self-adjoint indeed? More Precisely, Let $A:D(A)(\subset H)\to H$ a densely ...
0
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0answers
32 views

If $y_m \to y$ in $H$ then $|y_m|_H \leq C|y|$ for this sequence?

Let $w_i$ be a basis for a Banach space $V$. We have $V \subset H$ a continuous and dense embedding into a Hilbert space $H$. Define $y_m = \sum_{i=1}^m a_{im}w_i$. We have that $y_m \to y$ in a $H$ ...
35
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1answer
533 views

Over ZF, does “every Hilbert space have a basis” imply AC?

I know there is a similar result due to Blass [1] that over ZF, "every vector space has a (Hamel) basis" implies AC. Looking around, however, I can't find any results on the question for Hilbert ...
1
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1answer
161 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
1
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1answer
263 views

Can we have infinite-dimensional separable Hilbert spaces?

In Wikipedia's Hilbert space article on separability, it says: A Hilbert space is separable if and only if it admits a countable orthonormal basis. All infinite-dimensional separable Hilbert ...
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1answer
29 views

The projection operator defined by $(P_n(h) - h, v)_H = 0$

Let $V \subset H$ be separable Hilbert spaces with continuous embedding and suppose $\{v_n\}$ be a (non-orthogonal) basis for $V$. If we let $V_n = \text{span}(v_1, ..., v_n)$ and given $h \in H$ we ...
0
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1answer
82 views

Spectral Measures: Integration

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. How to define the integral for unbounded measurable functions: ...
3
votes
1answer
71 views

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is ...
1
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1answer
83 views

Subalgebra generated by selfadjoint operator $A_0\in\mathscr{L}(H,H)$

Let $\mathscr{L}(H,H)$ be the Banach algebra of bounded operators defined on a complex Hilbert space $H$ and let $B(A_0)$ be the subalgebra generated by the selfadjoint operator $A_0$, i.e. ...
3
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0answers
68 views

Sum of closed linear subspaces necessarily closed?

Let $H$ be an infinite-dimensional Hilbert space. Let $L_1,L_2 \subset H$ be two closed linear subspaces. If it is also known that $L_1 \perp L_2$ then it is not hard to show that $L_1 + L_2 = \{x_1 ...
1
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1answer
62 views

Why is the infinite dimensional vector space with only finitely many nonvanishing components incomplete?

Define a complex vector space $V$ such that any element $\{a_i\}=(a_1,a_2,\dots)\in V$ has only finitely many components $a_i\ne 0$. The inner product is defined as $$(\{a_i\},\{b_j\})=\sum_i^\infty ...
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0answers
99 views

Bounded operators with infinite matrix representations

Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and that $I$ is a non-empty set. If $A\subseteq B(K)$ for some Hilbert space $K$, we can ...
0
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1answer
29 views

A basis of $V$ is linearly independent in $H$ where $V \subset H$ are Hilbert spaces?

Let $V \subset H$ both be separable Hilbert spaces with continuous and dense embedding. Let $\{v_j\}$ be a basis for $V$, so every $v \in V$ can be written as $v = \sum_{j=1}^\infty a_jv_j$ with ...
2
votes
1answer
97 views

Hilbert Spaces - an application of the minimax principle.

Let $A$ be a compact, self-adjoint operator, $A \geq 0$. We need to prove that for any orthonormal system $\{e_i\}_1^{\infty}$ and for any $N$, $$\sum_1^N \langle Ae_i,e_i \rangle \leq \sum_1^N ...
6
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0answers
56 views

Two Hilbert spaces $V \subset H$, a basis for both spaces?

Let $V \subset H$ be a pair of Hilbert spaces (with different inner products). The embedding is continuous and dense, and both spaces are separable. Is it always the case that one can I find a ...
1
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1answer
84 views

Hilbert Spaces; eigenvalues of $PBP$ vs. $B$ for $B$ compact selfadjoint and $P$ orthoprojection.

An exercise I have come upon while studying Hilbert Spaces: Let $A$ be a compact operator, and $P \in L(H)$ be an orthoprojection. Prove that $$\lambda_n (PA^*AP) \leq \lambda_n (A^*A)$$ (Where ...
0
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3answers
22 views

Describing a Subset of a Hilbert Space $H$

Let $H$ be a Hilbert space. How can we describe the set $\{ x \in H \mid \|x-y\| = a \|x-z\| \},$ where $y, z \in H$ are fixed and $a > 0$? Geometrically how does it look like?
2
votes
2answers
89 views

Spectral Measures: Concentration

Given a Hilbert space $\mathcal{H}$. Consider spectral measures: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H}):\quad E(\mathbb{C})=1$$ Define its support: ...
0
votes
2answers
63 views

Spectral Measures: Property

Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$$ Can you give me a hint for: $$E(A)E(B)=E(A\cap B)$$ So far for disjoints I checked: ...
1
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1answer
80 views

Spectral Measures: Integration of Product

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. Define the integral of simple functions by: $$\int_\Omega ...
2
votes
1answer
177 views

Why are “not bounded” operators not everywhere defined?

Let $X, Y$ be Banach spaces, $\mathcal{D}(T)$ a subspace of $X$, and $T\colon X\to Y$ a linear map. Such a $T$ is commonly called an unbounded linear operator, where unbounded just means that the ...
2
votes
1answer
68 views

Estimating the modulus of continuity of translation in $L^2$ by a Sobolev norm of the function

For any $s\in \mathbb{R}$ define the Hilbert space $H^s(\mathbb{T})$ by means of norm $$\|f\|^2_{H^s}=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ Show that for any $0\leq ...
4
votes
2answers
129 views

left regular representation of SU(2)

in Sepanski's book Compact Lie groups, he describes the representation theory of SU(2) as being isomorphic to $\mathbb{N}$ (SU(2) acts irreducibly on the (n+1)-dimensional space of homogeneous ...
0
votes
1answer
19 views

Integration over subsets of the complex plane.

Original Problem: Let $\Omega\subset \mathbb{C}$ be an open set and let $f:\Omega\to\mathbb{C}$ be holomorphic such that $f\in L^{2}(\Omega)$. Show that if $B(z,r)$, the ball of radius $r$ ...
0
votes
1answer
51 views

Existence of minimum norm solution to linear equation $Tx =y$

Let $T: X \to Y$ be a bounded linear map between Hilbert spaces $(X, \langle \cdot , \cdot \rangle_X)$ and $(Y, \langle \cdot , \cdot \rangle_Y)$ (the Hilbert spaces may be complex or just real ...
2
votes
2answers
140 views

Exercise 23 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

Consider exercise 23 from chapter 4 ("Hilbert Spaces: An Introduction") of [1] (p. 198). Any help will be much appreciated. Thank you in advance. Suppose $\{T_k\}$ is a collection of bounded ...
1
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1answer
29 views

Extending mappings on simple tensors

Consider the following situation: Let $H, K$ be Hilbert spaces and let $\Phi$ be some mapping defined on simple tensors in $H\otimes K$ taking values in $B(H\otimes K)$ with the property that each ...
0
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1answer
77 views

Inner Product in Hilbert Space

Let $H$ be a Hilbert space and $\phi_{1}, \dots, \phi_{n} \in H$ are linearly independent vectors. How can we construct the inner product on $H$ such that $\phi_{1}, \dots, \phi_{n}$ become orthogonal ...
2
votes
1answer
286 views

Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
1
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1answer
41 views

Are there any interesting Hilbert spaces that do not present as function spaces?

I was pondering this question in class earlier: All separable, infinite dimensional Hilbert spaces are isometrically isomorphic. Thus, in particular, any such space is isometrically isomorphic to ...