# Tagged Questions

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### 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 ...
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### 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 $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $H$ 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 ...
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### 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 ...
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### 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?
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### 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} & ...
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### Linear algebra in Hilbert space

Let $M,N$ be closed subspaces of a separable Hilbert space. How to prove rigorously the following: $\operatorname{dim} M >\operatorname{dim} N => \exists u\neq0 \in M, u\in N^\perp$ ...
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### 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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### 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.
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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 $g \in ... 1answer 172 views ### Normal$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$? 0answers 106 views ### Can a Hermitian operator on a tensor product space be represented as a sum of tensor products of Hermitian operators? Consider a Hilbert space (or just a vector space over$\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 ... 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 $... 1answer 429 views ### Vector space generated by the tensor products of pauli matrices Let$\sigma_0,\sigma_x,\sigma_y,\sigma_z$stand for the$2\times 2$identity matrix and the well known pauli matrices: ... 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 ... 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$\mathcal{L}^2(\mathbb{R})$... 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.