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

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Books to learn tensor product on hilbert spaces

I have just started to work on Quantum Computing. I have began to read a paper which deals with tensor product on hilbert spaces. I have a had a course in functional analysis. So I don't have an ...
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Orthonormal Bases in a Hilbert Space and Vector Subspaces

Let $H$ be a Hilbert space and $S$ a vector subspace of $H$ which is dense in H. Does there exist an orthonormal basis $(u_{\alpha})_{\alpha \in A}$ of $H$ such that $\{ u_{\alpha} : \alpha \in A ...
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34 views

Smallest closed subspace of $A$ in pre-Hilbert spaces [duplicate]

Let be $A\subset H$ a subset of $H$ Hilbert space. I know that $A^{\perp\perp}$ is the smallest closed subspace of $H$, such that $A\subset A^{\perp\perp}$. But if $H$ is a inner product space (or ...
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73 views

$L^{2}[-\pi,\pi]$ is unitarily isomorphic to $l^2(\Bbb C)$

So I have countable orthonormal basis of $L^2[-\pi,\pi]$ as $\{e^{inx}\}_{n \in \Bbb Z}$ and countable orthonormal basis of $l^2(\Bbb C)$ as $\{a_n\}_{n \in \Bbb Z}$ such that ...
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61 views

Lie Algebra SU(2)

Given a two dimensional Hilbert-space, $\mathcal{H}$, and a vector $\eta \in \mathcal{H}$, of this space, if $\eta$ transforms in SU(2) like this, $$\eta \rightarrow e^{(-i\alpha ...
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28 views

Countable Complete Orthonormal Set implies countable dense subet

Let $\mathcal H$ be a Hilbert Space, let $B = \{u_j\}_{j=1}^{\infty}$ be a countable orthonormal basis. So we know that if a set is a complete orthonormal basis, the set of all finite linear ...
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131 views

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$. What I need is a verification and guidance. I managed to show that the set is orthogonal. My ...
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32 views

Use Cauchy-Schwarz inequality to prove that $\langle\,,\rangle : \mathscr H \times \mathscr H \to \Bbb C$ is continuous.

Let $(a,b) \in \mathscr H \times \mathscr H$ be fixed. So we have to prove that for a given $\epsilon \gt 0$, we can find $\delta_1 \gt 0$ and $\delta_2 \gt 0$ such that $\lvert \langle x,y\rangle - ...
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34 views

Prove that $f(x,y)$ defines an inner product [duplicate]

Let $(E,\left\lVert . \right\rVert)$ be a normed vector space defined on $\mathbb{R}$ . We suppose that the norm satisfies the Parallelogram law. Prove that: $$f(x,y)=(1/4)[(\left\lVert x+y ...
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18 views

Is the distance of an element $a$ from a subspace $M$ always $||a-P_M a||$?

The distance of an element $a$ from a subspace $M$ is $||a-P_Ma||$? ($P_Ma$ is the orthogonal projection of $a$ on $M$). During the course of studying about Hilbert Spaces and The Operators Theory, I ...
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52 views

Dense subsets of functional spaces

In books on Malliavin calculus and stochastic PDE, I found the following result is frequently used. I state it here in the simplest form. Given a separable Hilbert space $\left(H, \langle \cdot, ...
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36 views

Multiplication of Matrices in a Hilbert Space

So I was having a discussion with a friend as follows: Let $\mathcal H$ be a Hilbert space. Let $\mathcal H^{\otimes n} = \mathcal H \otimes \mathcal H \otimes \cdots \otimes \mathcal H$. $\mathcal ...
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2answers
45 views

Example of an operator that is not subnormal

In some recent questions the term subnormal operator has appeared. A bounded operator $A$ acting on a Hilbert space $H$ is called subnormal if there exists a Hilbert space $K$ containing $H$ as a ...
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23 views

Polarization of quadratic form yeilds sesquilinear form

How does polarisation of any quadratic form $Q: V \rightarrow \mathbb{C}$ on a complex vector space $V$ yields a sesquilinear form?
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19 views

Find a vector on H Hilbert separable [duplicate]

Im stuck with this excercise... Let H be and infinite-dimensional separable Hilbert space with $\{e_n\}_{n=1}^{+\infty}$ a Hilbert base. Given $\epsilon > 0$, find a vector $x_{\epsilon}\in$ ...
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Different versions of Mercer's theorem

I am reviewing materials on reproducing kernel Hilbert space (RKHS) and I've found various versions of Mercer's theorem: About the positive-definiteness conditions. In the Wikipedia pages on RKHS ...
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56 views

What does it exactly mean that finite linear combination to be dense?

This phrase comes up over and over again when studying Hilbert space, and since I don't have the strongest background in linear algebra, the statement like "finite linear combination of elements in ...
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37 views

For a normal operator $T$ we have $\sup_{\Vert x \Vert = 1} \mathrm{Re} \langle x, Tx \rangle = \sup_{\lambda \in \sigma(T)} \mathrm{Re} \lambda$

If $(X, \langle \cdot, \cdot \rangle)$ is a complex Hilbert-space and $T : X \rightarrow X$ a normal operator, i.e. an operator such that $T T^\ast = T^* T$ then I'd like to show that: ...
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27 views

Sequence of operators that commute imply the limit commutes?

Given a sequence of compact operators $A_n\to A$ as $n\ \to \infty$ and $B$ (which has finite rank). $\varphi \in L^2([a,b])$ If $A_nB\varphi = BA_n \varphi$ Am I able to say anything about $A$, ...
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38 views

Proof of $\hat{\mathrm{O}}$ta's theorem

I'm trying to prove $\hat{\mathrm{O}}$ta's theorem : Let $A$ be a closed operator on a Hilbert space $H$ and $\overline{\mathcal{D}(A)}=H$. Suppose that $A\mathcal{D}(A)\subset \mathcal{D}(A)$ and ...
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103 views

Classification of representations of compact $C^*$ algebras for single operators.

I am looking at Arveson's book, an invitation to $C^*$ algebras. There, it is explained p. 21 ($C^*$ algebras of compact operators) that any representation of a compact $C^*$ algebra can be decomposed ...
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39 views

An exercise about variational principle

Let $H$ be a Hilbert space. Let $l: H \to \mathbb{R}$ be a continuously linear function. Let $g: H \to \mathbb{R}$ be defined by $$g \left ( x \right )= \frac{\left \| x \right \|^2}{2}-l\left ( x ...
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25 views

Cartesian Decomposition.

I just read this on some notes written by my professor. It requires $X$ to be a linear map from complex Hilbert space $\mathcal{H}$ to itself, and that the Cartesian decomposition of $X$ is $X = H + ...
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Writing matrix representation of multiplication operator

For a given $m(x)\in L^2(0,1)$, let's write the multiplication operator $M\colon L^2(0,1)\longrightarrow L^2(0,1)$ as $Mf(x)=m(x)f(x)$. To write the matrix representation of this operator we need a ...
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37 views

Uniform convergence in Mercer Theorem for bounded kernels

Let $\mu$ be a finite, strictly positive measure on $\mathbb{R}$, and let $k$ be a measurable positive-definite kernel. Assume $k$ is bounded, and let $T:L^2(\mu)\rightarrow L^2(\mu)$ be defined by $$ ...
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31 views

GNS construction of a weight

In the theory of quantum groups in the operator algebraic setting, one deals with weights (instead of positive linear functionals). Definition: A weight is a function $\phi $ : $A^+ \rightarrow [0, ...
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20 views

convergence of series in inner product space

let $V$ be some inner product space and $\lbrace {e_i\rbrace }_{i\in\mathbb{N}} \subset V$ be some countable orthonormal set. I am wondering if for any $x\in V$ the series $$\sum\limits_{i=1}^{\infty} ...
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39 views

Spectral theory of compact, self-adjoint operators.

Let $T$ be a compact, self-adjoint operator on a separable Hilbert space H. Suppose that $f\in H$, $||f|| =1$ and $||(T-3)f||\leq 1/2$. Let P be the orthogonal projection onto the direct sum of all ...
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16 views

Initial and final sub-spaces of a partial isometry

Let $H$ be a Hilbert space and assume $H_0$ and $K_0$ are two sub-spaces of $H$ with dim$H_0$=dim$K_0$. Question: Is there any partial isometry $u$ whose initial projection is $H_0$ and final ...
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Finding the closest function to another in a Hilbert space.

Let H be the Hilbert space L$^2$([0,1)], and let $S$ be the subspace of functions f $\in$ H satisfying $\int^1_0(1+x)f(x)dx=0$. Find the element of $S$ closest to the function $g\in H$ defined ...
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30 views

Product of Hilbert bases of $L^2(\mathbb{R}^p)$ and $L^2(\mathbb{R}^q)$ is a Hilbert basis for $L^2(\mathbb{R}^{p+q})$

Let $(\alpha_n)_n$ be a Hilbert basis of $L^2(\mathbb{R}^p)$ and let $(\beta_k)_k$ be a Hilbert basis for $L^2(\mathbb{R}^q)$. I need to show that $(\alpha_n \beta_k)_{(n,k) \in \mathbb{Z}}$ is also a ...
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Linear span in the intersection of Hilbert spaces

Let $V$ be a vector space. Assume $H_1$ and $H_2$ are subspaces of $V$, and that both $H_1$ and $H_2$ are Hilbert spaces with inner-products $\langle \cdot, \cdot\rangle_1$ and $\langle ...
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47 views

Showing that a Hilbert Basis $(e_n)_{n \in \mathbb{N}}$ verifies $u= \sum (u,e_n)e_n $

The definition I have been given for a Hilbert Basis in a Hilbert Space $H$ over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ is: A sequence $(e_n)_{n \in \mathbb{N}}$ is an orthonormal basis if it ...
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Let $H$ be a Hilbert space and $Φ≤H$ be equipped with a topology. Under which topology on $Φ^*$ is $H^*\ni f\mapsto\left.f\right|_Φ$ continuous?

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $\Phi$ be a vector subspace of $H$ equipped ...
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36 views

Bilinear forms defines inner product on Hilbert Space

I have difficulties understanding the reason why when I have a self adjoint linear operator $T : \mathcal{H} \rightarrow \mathcal{H}$, and know that $A\|f\|^2 \leq \langle Tf,f \rangle \leq B\|f\|^2$ ...
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32 views

Proof of Anti-Linearity of Hermitian Conjugate

How can I prove that the adjoint operation/ Hermitian conjugate in anti-linear i.e $(\sum_{i} a_i A_i)^\dagger = \sum_{i} a_i^* A_i^\dagger$, where $A$ is any linear operator on a Hilbert space $V$. ...
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35 views

Is the span of a compact, connected and infinite dimensional subset of $l^2$ open as a subset of the closed span?

Let $K$ be a compact, connected and infinite dimensional subset of $l^2$. Is it possible to prove that $\operatorname{span}(K) \setminus \{0\}$ is open in $\overline{\operatorname{span}(K)}$?
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58 views

$5$ questions on the definition of the Gelfand triple

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb F\in\left\{\mathbb R,\mathbb C\right\}$, $\left\|\;\cdot\;\right\|$ be the norm induced by ...
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Show that the linear functional $\delta_{\lambda}(\varphi)=\varphi(\lambda),\delta: H^2(\Bbb{D})\to\Bbb{R}$ is continuous

Show that the linear functional $\delta_{\lambda}(\varphi)=\varphi(\lambda),\delta: H^2(\Bbb{D})\to\Bbb{R}$ is continuous where $\delta\in \Bbb{D}$, $\Bbb{D}$ is the unit disk, $H^2({\Bbb{D}})$ is a ...
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43 views

Do eigenvectors with pairwise distinct eigenvalues of a bounded, linear, nonnegative, symmetric operator on a Hilbert space build an orthogonal basis?

Let $H$ be a Hilbert space and $Q$ be a bounded, linear, nonnegative and symmetric operator on $H$ with finite trace. By the Hilbert–Schmidt theorem, there is an orthonormal basis ...
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32 views

Is $\delta:C_2[0,1]\to \Bbb{R}, \delta(f)=f(0)$ discontinuous?

Is the functional $\delta:C_{2}[-1,1]\to \Bbb{R}, \delta(f)=f(0)$ with $L_2$ norm discontinuous? I understood I should show whether or not the functional is unbounded, but I can't seem to understand ...
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Is there a condition along with Strong convergence which gives convergence in norm?

Let $\{T_n\}$ be a sequence of operators in a Hilbert Space and $T_n$ converges to an operator $T$ in the strong operator topology. Is there any assumption which I can put on $\{T_n\}$ so that the SOT ...
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25 views

If $(u_n)_n,(v_n)_n$ are seq. in a Hilbert space and $(e_k)_k$ is an ONB, then $\sum_k|\sum_n(u_n,e_k)(v_n,e_k)|≤\sum_n\sum_k|(u_n,e_k)(v_n,e_k)|$

Let $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $(u_n)_{n\in\mathbb N}\subseteq H$ and ...
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37 views

The vector-valued distribution of compact support

Let $\mathcal{H}$ be infinite dimensional Hilbert space and $D(\mathbb{R}^n)$ be the space of smooth complex functions of compact support. Consider the distribution $T: D(\mathbb{R}^n) \to ...
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53 views

Functional analysis - check that a closed subspace of a Hilbert space is convex

Suppose that V is a Hilbert space over $F$ and $W$ is a closed subspace of $V$ . Then for every $x \in V$ , there exist unique $y \in W$ and $z \in$ (the orthogonal compliment of $W$) such that $x = y ...
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21 views

Are these two statements involving inf and sup equivalent?

Suppose $H$ is a Hilbert space. Is $$\inf_{h \in H}\sup_{g \in H}\frac{f(h,g)}{|h||g|} \geq C$$ the same as $$f(h,h) \geq C|h|^2\quad \forall h \in H$$ ? Here $f\colon H \times H \to ...
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57 views

Show that there exists a certain operator in $L(H)$ where $H$ is a separable Hilbert Space.

Given a separable Hilbert Space $H$ and $\sum_{n=1}^{\infty} f_n$ an absolutely convergent series in $H$, I need to show that there is an operator $A \in L(H)$ such that $A(e_n)=f_n$, where ...
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134 views

Vector space that can be made into a Banach space but not a Hilbert space

Are there any (real or complex) vector spaces which can be made into a Banach space given a suitable norm, but cannot be given a norm that makes it a Hilbert space? I know that the parallelogram law ...
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2answers
79 views

Polynomials form a Hilbert basis for $L^2$

If you form a set of orthonormal polynomials on $[0,1]$, by applying the Gram-Schmidt process from monomials $\{1, x, x^2, \dots \}$ then what is required to show that this is a basis for $L^2[0,1]$? ...
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71 views

Comparing two sigma algebras in Hilbert spaces

Let $H$ be a non-separable Hilbert space. We denote $B$ by the sigma algebra generated by the norm topology in $H$. We also denote $B_{w}$ by the sigma algebra generated by the weak topology in $H$. ...