Questions tagged [hilbert-spaces]

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

Let $H$ a vector space over the field $\mathbb C$, and $\langle \cdot,\cdot\rangle\colon H\times H\to \mathbb{C}$ a map which satisfies

  1. $\langle x,x\rangle =0\Longrightarrow x=0$ and $\langle x,x\rangle\geqslant 0$ for all $x\in H$,
  2. $(\forall x,y\in H):\langle x,y\rangle=\overline{\langle y,x\rangle}$,
  3. $(\forall x_1,x_2,y\in H)(\forall\alpha_1,\alpha_2\in\mathbb C):\langle \alpha_1 x_1+\alpha_2 x_2,y\rangle=\alpha_1\langle x_1,y\rangle+\alpha_2\langle x_2,y\rangle$.

The map $\lVert\cdot\rVert\colon H\to\mathbb R_+$, defined by $\lVert x\rVert =\langle x,x\rangle^{\frac 12}$ is a norm.

If $(H,\lVert \cdot\rVert)$ is complete, then $H$ is called a Hilbert space.

Example: The space $H$ of all sequences $x_0,x_1,x_2,\ldots$ of complex numbers such that $\sum_{n=0}^\infty|x_n|^2<\infty$, with the inner product $$\bigl\langle(x_0,x_1,x_2,\ldots),(y_0,y_1,y_2,\ldots)\bigr\rangle =\sum_{n=0}^{+\infty}x_n\overline{y_n}$$is a Hilbert space.

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What is "Bra" and "Ket" notation and how does it relate to Hilbert spaces?

This is my first semester of quantum mechanics and higher mathematics and I am completely lost. I have tried to find help at my university, browsed similar questions on this site, looked at my textbook (Griffiths) and read countless of pdf's on the…
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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 space} & \text{complete space}\\ \text{norm} & \text{normed} &…
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What do mathematicians mean by "equipped"?

I am a mathematical illiterate, so I do not know what people mean when they say "equipped". For example, I say that a Hilbert space is a vector space equipped with an inner product. What does that actually mean? Obviously, one interpretation is to…
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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 really grok this other linear transformation…
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Separable Hilbert space have a countable orthonormal basis

I want to show that every an infinite-dimensional separable (contains countable dense set) Hilbert space has a countable orthonormal basis. I know that every orthogonal set in a separable Hilbert space is countable, it is help me with the proof?
ali baba
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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 spaces. I also don't see how to generalize Blass's…
Kameryn Williams
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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 $T$? This operator doesn't seem to be an orthogonal…
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What is the difference between isometric and unitary operators on a Hilbert space?

It seems that both isometric and unitary operators on a Hilbert space have the following property: $U^*U = I$ ($U$ is an operator and $I$ is an identity operator, $^*$ is a binary operation.) What is the difference between isometry and unitary?…
3 answers

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 $\sum_{n=1}^\infty b_n^2<\infty$, then $\sum_{n=1}^\infty a_n^2…
Bruno Stonek
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$A$ and $B$ commute on a dense set but $e^{iA}$ and $e^{iB}$ do not

Let $A$ and $B$ be unbounded, symmetric operators on a Hilbert space $H$ with a common domain $D$. If $AB = BA$ on $D$, is it necessarily that case that $e^{iA}$ and $e^{iB}$ also commute? If $A$ and $B$ are bounded, then I know that this must be…
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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: $$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ \text{for all} \ 1\le j\le N?$$ Notes and…
Carl Brannen
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$\ell_p$ is Hilbert if and only if $p=2$

Can anybody please help me to prove this: Let $p$ be greater than or equal to $1$. Show that for the space $\ell_p=\{(u_n):\sum_{n=1}^\infty |u_n|^p<\infty\}$ of all $p$-summable sequences (with norm $||u||_p=\sqrt[p]{\sum_{n=1}^\infty |u_n|^p}\…
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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 vector can be written as a finite sum of vectors from…
Gadi A
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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?
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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 a unique point $p'$ in $K$ closest to $p$. We…
Mike F
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