1
vote
0answers
29 views

Linear map in Hilbert space.

If you have a linear map $h\mapsto T(h)$ from $H_1$ a real separable space, to Hilbert space $H_2$, it seem that this maps provides an isometry of $H_1$ onto a closed subspace of $H_2$. I try to ...
3
votes
1answer
51 views

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$.

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$. Let consider $H=l_2$ where $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace $ ...
1
vote
2answers
65 views

Hilbert vs Inner Product Space

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product. Also can a metric always be defined ...
5
votes
1answer
63 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
0
votes
1answer
58 views

Why is $L^2$ function Hilbert space not defined for Riemann Integral

The space of square Lebesgue integrable functions is said to be a Hilbert space. Why is if the integral is Riemann then this is not a Hilbert space? In other words, why not the space of Riemann square ...
1
vote
2answers
55 views

Chart of how the mathematical spaces are related? (soft question)

When dealing with specific function spaces e.g. Sobolev, Hilbert, etc., I find it easy enough to accept the properties of that space and work with them; however, I have a hard time visualizing how ...
0
votes
1answer
51 views

Proving Density of Subset of Hilbert Space

Suppose we have a subspace, $M$, of Hilbert space $H$. Prove the first statement implies the second statement: 1) If $<f,g> = 0$ for any $g\in M$, then $f=0$ in $H$. 2) $M$ is dense in $H$. I ...
2
votes
1answer
66 views

Understanding problems of space

I've been trying to understand the concept of space for some time now, but I still can't grasp the essence of it. In high school math we've been using 2D- and 3D- Euclidean space. Now that I am ...
0
votes
1answer
31 views

How to show that the limit of sequence of eigenvectors (same eigenvalue) is also an eigenvector?

Let $H$ be a continuous Hermitian operator on an infinite dimensional Hilbert space. Also, let $f_n$ be a sequence approaching $f$ as $n\to\infty$, where each $f_n$ is an eigenvector of the same ...
0
votes
4answers
100 views

Why is the inner product not an element of the Hilbert space?

What I know about Hilbert space is that, elements in that space can be complex numbers. But I was confused to read this statement from a book: The inner product, being a complex number, is not an ...
3
votes
1answer
102 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
0
votes
1answer
36 views

$H$ Hilbert, $\ker L \neq H \Rightarrow (\ker L )^{\perp} \neq \lbrace 0 \rbrace$

If $H$ is a Hilbert space on $\mathbb{C}, L : H \rightarrow \mathbb{C} $ is linear and bounded, $\ker L \neq H $ then $ (\ker L )^{\perp} \neq \lbrace 0 \rbrace.$ It seems like a quite easy ...
0
votes
2answers
35 views

Representation of a vector

$(l^2,\|\cdot\|_2)$ is a Hilbert space with scalar product $\langle x,y\rangle=\sum^{\infty}_{k=1}x_ky_k$. How can I show that every vector $x\in l^2$ can be written in a form ...
1
vote
0answers
89 views

Nice tutorial for Reproducible kernel hilbert space

I am looking for some nice tutorials related to Kernel Hilbert space. I went through lot of them but I couldn't figure out even why it is called reproducible. Any suggestions guys?
0
votes
0answers
108 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} & ...
1
vote
2answers
79 views

Linear algebra in Hilbert space

Let $M,N$ be closed subspaces of a separable Hilbert space. How to prove rigorously the following: $\operatorname{dim} M >\operatorname{dim} N => \exists u\neq0 \in M, u\in N^\perp$ ...
3
votes
2answers
58 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
1answer
264 views

How can I able to show that $(S ^{\perp})^{\perp}$ is a finite dimensional vector space.

Let $H$ be a Hilbert space and $S\subseteq H$ be a finite subset. How can I able to show that $(S ^{\perp})^{\perp}$ is a finite dimensional vector space.
2
votes
1answer
46 views

On the existence of a bounded linear functional

Let $\mathcal{H}$ be a Hilbert space. By the Riesz Representation Theorem, we have that any bounded linear $\psi \in \mathcal{H}^{*}$ is of the form $\psi(h) = \langle h, g \rangle$ for some $g \in ...
4
votes
1answer
172 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
1
vote
0answers
106 views

Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators?

Consider a Hilbert space (or just a vector space over $\mathbb{C}$), which is a tensor product of several smaller Hilbert spaces: $$ H = H_1 \otimes \cdots \otimes H_n, $$ and let $\mathcal{H}$ be a ...
4
votes
5answers
774 views

Subspaces of Hilbert Spaces of finite dimension

Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ ...
0
votes
1answer
429 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
0
votes
1answer
49 views

Dimension of a set and its closure are equal in an Inner-product space?

I want to show that, given a subset $M$ of an Inner Product space $X$. If $M$ is a total set then, $M^\perp=\{0\}$. Which I have shown using the completion of $X$, which will be a Hilbert Space. And ...
1
vote
2answers
476 views

Does the vector space spanned by a set of orthogonal basis contains the basis vectors themselves always?

I used to think that in any Vector space the space spanned by a set of orthogonal basis vectors contains the basis vectors themselves. But when I consider the vector space $\mathcal{L}^2(\mathbb{R})$ ...
4
votes
1answer
311 views

Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces? For the sum we have the notion of a direct integral, here.
2
votes
3answers
1k views

Canonical examples of inner product spaces that are not Hilbert spaces?

That is, what are some good examples of vector spaces which are inner product spaces but in which not every Cauchy sequence converges?
23
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 ...
7
votes
1answer
951 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?
25
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\;\;\; \ ...