# Tagged Questions

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

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### Question on operator of hilbert space, why $f(x)=\sum_{i}(f|e_i)e_i$?

let $(V,(. |.))$ a Hilbert space. Let $\{e_i\}_{i=1}^\infty$ an orthonormal basis and $f:V\to V$ a linear application. Here are my questions : 1) Why $f(x)=\sum_{i=1}^\infty (f,e_i)e_i$ ? 2) Why ...
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### What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
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### Showing the basis of a Hilbert Space have the same cardinality

I am trying to show that if we have two orthonormal families $\{a_i\}_{i\in K}$ and $\{b_j\}_{j\in S}$ and these are the basis of some Hilbert Space H, then they have the same cardinality. So If I ...
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### Quantum Mechanics: position and the separability of Hilbert space?

I would be pleased if someone could point out to me where I go wrong in the following sequence of statements: One model of quantum mechanics identifies states of a particle with normalized vectors (...
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### What is the relation between the matrix of a bounded linear operator and that of its adjoint operator?

Let $H_1$ and $H_2$ be finite-dimensional (real or complex) Hilbert spaces, let $T \colon H_1 \to H_2$ be a linear operator, [Then $T$ can be shown to be bounded] and let $T^* \colon H_2 \to H_1$ ...
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### How to prove $n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$ is a norm on $B(H)$ and $n(T)\lt\|T\|\lt2n(T)$ where $T\in B(H)$? [closed]

Let $H$ be a Hilbert space over $\mathbb C$. If $T\in B(H)$, how to prove that $$n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$$ is a norm on $B(H)$ and $$n(T)\lt||T||\lt2n(T)\ \textrm{?}$$ I couldn'...
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### Quantum Mechanics state space

In Quantum Mechanics one often deals with wavefunctions of particles. In that case, it is natural to consider as the space of states the space $L^2(\mathbb{R}^3)$. On the other hand, on the book I'm ...
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### Is a Bessel sequence a frame sequence?

$\mathcal H$ being a Hilbert space, $\{g_k\}_{k \in N}$ is a Bessel sequence if there exsits $B >0$ such that $\forall f \in \mathcal H$, $\sum_{k\in N} |\langle f,g_k\rangle|^2 \leq B \| f \|^2$. ...
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### $H$ self-adjoint with mass gap, $P \ge 0,\Omega \in D(P)$, $H + \lambda P$ self-adjoint $\implies$ for $\lambda$ small, $H+ \lambda P$ has gap?

Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...
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### What does the weak* topology on $\ell_2$ look like?

I am wondering about a way to construct a base or subbase for the weak* topology on $\ell_2$. I am fairly new to topology and functional analysis, so I apologize if the question is not precisely ...
In my work, I encountered the following problem. Consider the set of real-valued functions, which are balanced'', that is the set of bounded functions $f(x)$ such that $\lim_{x\rightarrow \pm \... 0answers 43 views ### Show that usual$L^2$norm is equivalent to arbitrary norm$||\cdot||$that satisfies 'convergence condition' Given$L^2(\mathbb{R})$consider a norm$||\cdot||$on$L^2$such that$(L^2,||\cdot||)$is a Banach space and every$||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ... 1answer 25 views ### Prove or disprove:$\lVert T\rVert=\sup_{\lVert x\rVert=1}|\langle Tx,x\rangle|$, where$H$is a Hilbert space and$T$is bdd linear operator. Edit: To clarify, note that$T:H\to H$. This is a problem on an old preliminary exam in Analysis that I'm working through to prep for my own prelim. My initial thought was to disprove it, but I can'... 0answers 55 views ### Generalized Absolute Value II Let$x$be an operator in$B(H)$. We say a pair$(c,y)$forms a polar decomposition for$x$if$y$is a positive operator,$c$in$B(H)$with$x=cy$such that the restriction of$c$on$\overline{yH}$... 0answers 39 views ### Example of Hilbert space non isomorphic to$L2$I'm looking for an example of a Hilbert space that can't be seen as the countable direct sum of$L^{2}(X,\mu)$spaces nor subespaces of$L^{2}(X,\mu)$. Some idea to start? Thanks everyone. 1answer 2k views ### The difference between an isometric operator and a unitary operator on a Hilbert space? It seems that both isometric and unitary operators on a Hilbert space have the following property:$U^*U = I$($U$is an operator and$I$is the identity operator) What is the difference between ... 1answer 17 views ### Sesquilinear forms - How does positiveness imply hermitianity? In my mathematical methods for physics course notes I find this: A positive sesquilinear form is nondegenerate and Hermitian The first statement is trivial: a ... 0answers 31 views ### Completeness of 'Hardy Space'$H^2(D)$Define Hardy Space$H^2(D)$as a space of holomorphic functions$f$on unit open disc$D=\{z\in\mathbb{C}:|z|<1\}$endowed with the norm $$||f||^2=\sup_{0<r<1} \int_0^{2\pi} |f(re^{i\... 1answer 34 views ### Spectrum of two Hilbert spaces Let H_1 and H_2 be two Hilbert spaces and U \in B(H_1,H_2) be unitary. Assume that A\in B(H_2) and B \in B(H_1) satisfy UB = AU. How can I prove that sp(A) = sp(B) and sp_p(A) = sp_p(B)?... 0answers 13 views ### Application of Uniform boundedness theorem: \langle Tx,y\rangle bounded for each x,y then ||T|| is bounded For Hilbert Space X, if we have a condition on a subset F\subset B(X) ('set of bounded linear operators on X') such that$$ \{\langle Tx,y\rangle:T\in F\} $$is a bounded set for each x,y\in ... 1answer 22 views ### Stuck on elementary proof on completeness of W^{1,2}(\mathbb{R}) as Hilbert Space Let W^{1,2}(\mathbb{R}):=X be the space of continuous functions f such that f\in L^2(\mathbb{R}) and there exists f'\in L^2(\mathbb{R}) such that$$ f(b)-f(a)=\int_a^b f'(t)\,dt $$for ... 1answer 23 views ### Characterization of orthogonal projections in terms of operator norms I want to show the following equivalence: If X is a Hilbert Space and P\in B(X) (i.e. P is bounded and linear) and P^2=P, then$$ (\text{im}\,P)^{\perp} =\ker P\iff ||P||\le 1 $$I know that ... 0answers 30 views ### Hilbert Spaces and Hamel Basis Let H be a Hilbert Space of infinite dimension, S a not finite orthonormal basis and B a Hamel basis to H. i) How to show that the cardinality of B is greater than or equal to the ... 1answer 49 views ### Two orthonormal sets in a Hilbert space. One is complete, the other must be complete. Given two orthonormal sets \{e_k\}_{k=1,2\ldots}, \{e'_k\}_{k=1,2\ldots} in a Hilbert space H, which satisfy$$ \sum_{k=1}^\infty \|e_k-e'_k\|^2 < 1. \tag{*}$$Prove that if$\{e_k\}_{k=1,2\...
It is well known that using non-commutative Gelfand-Naimark theorem for finite dimensional $C^∗$-algebra we can obtain isometric representation on finite dimensional Hilbert space. My question is : ...