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

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34
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1answer
407 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 ...
28
votes
4answers
1k views

Given two basis sets for a finite Hilbert space, does an unbiased vector exist?

Let $\{A_n\}$ and $\{B_n\}$ be two bases for an $N$-dimensional Hilbert space. Does there exist a unit vector $V$ such that: $$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ ...
26
votes
2answers
3k views

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric ...
23
votes
3answers
815 views

If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$

I'm trying to prove the following: If $(a_n)$ is a sequence of positive numbers such that $\sum_{n=1}^\infty a_n b_n<\infty$ for all sequences of positive numbers $(b_n)$ such that ...
16
votes
2answers
4k views

Finding the adjoint of an operator

This is from my homework, I'm totally lost as to how to proceed. Consider the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined by $(Tf)(x) = \int^x_0 f(s) \ ds$ What is the adjoint of $T$? ...
16
votes
2answers
769 views

Is a closed set with the “unique nearest point” property convex?

A friend of mind had a question that I couldn't answer. It is well-known that if $K$ is a closed, convex subset of a Hilbert space $H$ (say over the reals) then, for any point $p \in H$, there exists ...
15
votes
1answer
702 views

Is there a constructive proof of this characterization of $\ell^2$?

I would like to revisit this question, which can be equivalently stated as: Proposition. Let $(a_n)$ be a sequence of real (or complex) numbers such that $\sum a_n b_n$ converges for every $(b_n) ...
15
votes
4answers
745 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 ...
14
votes
3answers
825 views

Intersection between orthogonal complement of a subspace and a set

Consider the normed vector space $E=\mathbb{R}^n$. Define $ P=\{x \in \mathbb{R}^n: x_i \geq 0, \forall i \}$. Let $M$ be a subspace such that $M \cap P = \{0\}$. I want to see that $M^\perp \cap ...
13
votes
1answer
1k views

How to prove that an operator is compact?

Consider $T\colon\ell^2\to\ell^2$ an operator such that $Te_k=\lambda_k e_k$ with $\lambda_k\to 0$ as $k \to \infty$ how to prove that it is compact?
13
votes
1answer
743 views

Different versions of Riesz Theorems

In Wikipedia, there are three versions of Riesz theorems: 1 The Hilbert space representation theorem for the (continuous) dual space of a Hilbert space; 2 The representation theorem for ...
12
votes
3answers
2k views

An orthonormal set cannot be a basis in an infinite dimension vector space?

I'm reading the Algebra book by Knapp and he mentions in passing that an orthonormal set in an infinite dimension vector space is "never large enough" to be a vector-space basis (i.e. that every ...
12
votes
4answers
365 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
12
votes
1answer
1k views

Operator norm and tensor norms

I have a linear operator $A\in\mathcal{L}(X,Y)$ where $X$ and $Y$ are some Banach spaces (or Hilbert spaces would also do, if that simplifies the answer.). The operator norm of $A$ is given by $$ ...
12
votes
1answer
656 views

Meaning of “kernel”

In analysis, there are at least three kinds of "kernel" concepts: In probability theory, there is a concept called transition probability, also called probability kernel, from one measure space $X$ ...
11
votes
2answers
2k views

The direct sum of two closed subspace is closed? (Hilbert space)

I know that if $X$ is a Banach space, then, the direct sum of two closed subspace $X_1$ and $X_2$ is not necessarily closed. But what if $X$ is Hilbert? I assume there is something to do with the ...
11
votes
2answers
143 views

Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...
11
votes
1answer
398 views

Quantization of angular momentum: is Dirac's proof wrong?

I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
10
votes
1answer
230 views

Criteria of compactness of an operator

Suppose $K$ is a linear operator in a separable Hilbert space $H$ such that for any Hilbert basis $\{e_i\}$ of $H$ we have $\lim_{i,j \to \infty} (Ke_i,e_j) = 0$. Is it true that $K$ is compact? ...
10
votes
1answer
470 views

orthonormal system in a Hilbert space

Let $\{e_n\}$ be an orthonormal basis for a Hilbert space $H$. Let $\{f_n\}$ be an orthonormal set in $H$ such that $\sum_{n=1}^{\infty}{\|f_n-e_n\|}<1$. How do I show that $\{f_n\}$ is also an ...
10
votes
1answer
561 views

Commuting operators and polar decomposition

Suppose that $V$ is an isometry and $X$ an arbitrary operator on a Hilbert space $H$. Let $X=U|X|$ be the polar decomposition for $X$. If $VX=XV$, can I conclude that $VU=UV$?
10
votes
1answer
347 views

Every Hilbert space operator is a combination of projections

I am reading a paper on Hilbert space operators, in which the authors used a surprising result Every $X\in\mathcal{B}(\mathcal{H})$ is a finite linear combination of orthogonal projections. The ...
9
votes
4answers
886 views

Measure on Hilbert Space

On $\mathbb{R}^n$, we of course have the usual Lebesgue meausre. In many ways, separable, infinite-dimesional Hilbert space is the most natural generalization of $\mathbb{R}^n$ to ...
9
votes
2answers
568 views

Proving an inequality with Cauchy-Schwarz

In the "User's guide to viscosity solutions" by Crandall, Ishii and Lions (link), they make the following claim (inequality (A.4) p. 58) : Given $x$, $\xi$ $\in \mathbb{R}^n$, $A \in \cal{S}(n)$ ...
9
votes
1answer
347 views

How to prove Halmos’s Inequality

How to prove Halmos’s Inequality? If $A$ and $B$ are bounded linear operators on a Hilbert space such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$ I found it from ...
9
votes
1answer
333 views

A paradox on Hilbert spaces and their duals

I am making some elementary mistakes here. Could you please help me point out the problems? Thank you very much! Suppose on some space $H$ we have two inner products, which make $H$ after completion ...
9
votes
3answers
210 views

Conjecture: the function $d(x, y):=\frac{||x-y||}{\max(||x||, ||y||)}$ is a distance

I make the following conjecture: the function $$ d(x, y):=\frac{||x-y||}{\max(||x||, ||y||)} $$ is a distance on $H$, where $H$ is a normed vector space or a Hilbert space, and $x, y \in H$ (the ...
9
votes
1answer
178 views

Is the identity map $id: H^2(-\pi,\pi) \to L^2(-\pi,\pi)$ Hilbert-Schmidt?

Let $H_1, H_2$ be Hilbert spaces. A linear operator $A: H_1 \to H_2$ is Hilbert-Schmidt iff for some orthonormal basis $\lbrace e_n : ~ n \in \mathbb{N} \rbrace$ of $H_1$ the sum $\sum_{n \in ...
9
votes
1answer
340 views

An approximate eigenvalue for $ T \in B(X) $.

This is a problem from Conway’s Functional Analysis: Definition An approximate eigenvalue for $ T \in B(X) $ is a scalar $ \lambda $ such that there is a sequence of unit vectors $ x_{n} \in X $ ...
8
votes
5answers
968 views

How to show that this set is compact in $\ell^2$

Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$ According to the book [Dunford and ...
8
votes
1answer
1k views

Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
8
votes
2answers
483 views

Haar's base for $L^2[0,1]$

$\newcommand{\span}{\operatorname{span}}$ 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}\\ ...
8
votes
2answers
489 views

Is compactness a stronger form of continuity?

Let $H$ be a Hilbert space. We say that a linear operator $T \colon H \to H$ is compact if it maps bounded sets to precompact ones, that is, if for every bounded sequence $(a_n)$ in $H$, $(Ta_n)$ has ...
8
votes
1answer
1k views

Equivalent inner products on a Hilbert space

Take a Hilbert space $(\mathcal H,(\cdot,\cdot)_{\mathcal H})$ and two equivalent inner products $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$ on $\mathcal H$, i.e. such that there are $a,b \in \mathbb R$ ...
8
votes
1answer
628 views

Transforming a distance function to a kernel

Fix a domain $X$: Let $d : X \times X \rightarrow \mathbb{R}$ be a distance function on $X$, with the properties $d(x,y) = 0 \iff x = y$ for all $x,y$ $d(x,y) = d(y,x)$ for all $x,y$ Optionally, ...
8
votes
1answer
775 views

Is the right shift operator bounded?

I was reading my lecture notes for functional analysis when I came across the following statement: Let $(e_{n})$ be a total orthonormal sequence in a separable Hilbert space H. The right shift ...
8
votes
1answer
411 views

Physical (Quantum Mechanical) Significance of completeness of Hilbert Spaces.

I'm not sure if the question is very 'mathematical',but I'm asking any way. I have a basic knowledge of quantum mechanics and I'm studying Hilbert spaces. I was wondering what is the physical ...
8
votes
1answer
560 views

What is the use of Spectral Theorem?

Obviously the version for compact and self-adjoint linear operators on Hilbert Spaces is very useful since it decomposes the operators into orthogonal projections. However, the following more general ...
8
votes
2answers
129 views

Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
8
votes
1answer
286 views

How to Prove the Semi-parametric Representer Theorem

This question concerns the generalized Representer Theorem, due to Schölkopf, Herbrich, and Smola. In this magnificent work, the authors provide two versions of the Representer Theorem, a ...
8
votes
1answer
234 views

Is my statistician friend right/wrong on metric spaces and norms?

I was talking to a statistician friend of mine who said that instead of minimizing this function $\sum_{i,j}W_{ij}d_{ij}^2(X)$ over $X$ it would be better to solve an analogous minimization problem ...
8
votes
0answers
371 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
7
votes
4answers
2k views

A linear operator on a finite dimensional Hilbert space is continuous

How do I show that a linear function from a Hilbert space $H$ to itself is continuous if $H$ is finite dimensional? Also, what would be an example of a linear function from a Hilbert space to itself ...
7
votes
1answer
522 views

Isometric to Dual implies Hilbertable?

Let $X$ be a Banach space and suppose that $X$ is isometric to its continuous dual space $X^*$. Must $X$ be hilbertable in the sense that there exists an inner product which induces the norm on $X$? ...
7
votes
4answers
610 views

An idempotent operator is compact if and only if it is of finite rank

Would you help me to solve this problem. Show that an idempotent operator on hilbert space is compact if and only if it has finite rank.
7
votes
2answers
2k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
7
votes
1answer
678 views

Does there exist a real Hilbert space with countably infinite dimension as a vector space over $\mathbb{R}$?

Essentially what the title says - where to me a Hilbert space is a complete (Hermitian) inner product space, am I safe to assume every such real Hilbert space is of uncountable dimension over ...
7
votes
1answer
321 views

Trace class for operators

Let $ \mathcal{H} $ be a Hilbert space and $ T: \mathcal{H} \to \mathcal{H} $ a bounded linear operator. The $ n $-th singular number $ {\mu_{n}}(T) $ of $ T $ is defined as the distance from $ T $ ...
7
votes
1answer
258 views

Existence of the Pettis integral

This is related to a question of MO: A question on the integral of Hilbert valued functions. I'm sure it's easy, but I cannot think right now, so I thought I'd ask. Let $f:[0,1]\rightarrow H$ be a ...
7
votes
1answer
228 views

Energy estimate of the differential equation $\dot{x}=Ax$

Conside the differential equation $$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...