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

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Sequence of unitary l.i. vectors such tha the sequence converges weakly to a non-zero vector, but not strongly

Let $\mathcal H$ be an infinite dimensional Hilbert space and let $\{x_{n}\}_{n=1}^{\infty}$ be a sequence of unitary linearly independent vectors. I know, using Bessel's inequality, that if the ...
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2answers
87 views

Square root of a bounded operator in Hilbert space

Consider the power series expansion $$ \sqrt{1-z} = 1+\sum\limits_{k=1}^\infty c_k z^k, $$ converging absolutely in the ball $|z| \leq 1$. Let $H$ be a Hilbert space and $A \in \mathcal L(H)$ a ...
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1k views

Prove that this integral operator is compact

Let $X,Y=L^2(0,1)$, $k\in C^0([0,1]^2)$. Define $$ K:X\to Y,\,\,\,\,\,Kf(x):=\int_0^1k(x,y)f(y)dy\,\,\,\,\forall\, f\in L^2(0,1). $$ I have to show that $K$ is compact. My idea is to prove that $K$ ...
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1answer
50 views

Matrix space, with $\langle A,B\rangle=\text{tr}(AB^*)$ isn't Hilbert space, how can i find a counter example?

Generally, I'm having quite troubles thinking about counter examples. So I would love if someone could guide me into finding the example for the following question by myself (and not just giving it ...
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1answer
43 views

Show that $\lbrace S_n x \rbrace$ converges for a particular recursively-defined sequence of operators $S_n$

$H$ is a Hilbert space, $M$ is a self-adjoint bounded linear operator on $H$ with $M \leq I$, and $S_0 = 0$; $S_{n+1} = (1/2)(M + S^2_n)$ for $n = 0, 1, 2, ...$. For all $n$, both $S_n$ and $S_n - ...
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1answer
29 views

Verifying a bound on the norm of an operator in $l_2$.

The problem: Define $L: l_2 \rightarrow l_2$ by $L(x_1, x_2, ...) = (y_1, y_2, ...)$, where $y_n = (x_1 + x_2 + ... + x_n)/n^2$. Show that $||L|| \leq (\sum_{n=1}^\infty 1/n^2)^{1/2}$. My proof: ...
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1answer
68 views

Spectral Measures: Domain Criterion

Problem Given a Hilbert space $\mathcal{H}$. Consider a spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Then the criterion holds: ...
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2answers
87 views

Hilbert space: product and tensor product space

Let $H_1$ and $H_2$ be Hilbert spaces, then I would intuitively define the inner product on $H_1 \times H_2$ by $\langle (x_1,x_2),(y_1,y_2)\rangle = \langle x_1,y_1 \rangle + \langle x_2,y_2 ...
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1answer
63 views

Correspondence between bounded sesquilinear forms and bounded linear operators

Let $H,K$ are Hilbert spaces, I want to show there is an isometric linear correspondence between bounded sesquilinear forms $S(H,K)$ and bounded linear operators $B(H,K)$. ( $\Phi: B(H,K)\to S(H,K)$ ...
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1answer
102 views

Spectral Measures: Square Root

Isometric Equality Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^{**}$$ Denote for shorthand: $$H:=A^*A:\quad H=H^*$$ Regard elements: ...
5
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1answer
69 views

Showing a certain subspace of Hilbert space is dense

Let H be the Hilbert space of square-summable sequences of reals. A few years ago I thought I had proved that the subspace Z of real sequences with only finitely many nonzero terms, such that they ...
3
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1answer
70 views

Different norm on $\ell_p$-space and Hilbert space

We define $\ell_p=\{(x_n)_{n\in{\mathbb{N}}}\in\mathbb{C}^\infty:\sum_n{|x_n|^p}<\infty\}$. With the usual usual norm $||.||_p$ this becomes a Bancach space. Also we have the usual inner product : ...
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1answer
69 views

Central Limit Theorem for transformed random variables

The Central limit theorem (CTL) is often given similar to the entry in Wikipedia as: Suppose ${X_1, X_2, ...}$ is a sequence of independent and identically distributed random variables with ...
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26 views

Coefficients in Representer theorem

I have a Mercer Kernel, $K\colon X \times X \rightarrow \mathbb{R}$, i.e. it is continuous, symmetric and postive definite on a compact domain $X \subset \mathbb{R}^n$. Also, I have a set of $m$ ...
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1answer
171 views

How are Hilbert Space methods used in number theory?

In N. Young's book An Introduction to Hilbert Space, there is an interlude in which the author remarks that the theory of Hilbert spaces is "routinely used in differential geometry, complex analysis, ...
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53 views

Basis of intersections of $L^p$ spaces

I keep confusing myself about a subspace basis and I can only find intelligible material discussing the finite, linear algebra, case. It is known that the Hilbert space $L^2(X)$ has a basis, for ...
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3answers
145 views

Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$.

Show that an isometric linear operator $T:H\to H$ satisfies $T^* T=I$, where $I$ is the identity operator on $H$. I've been stuck on this for a while and don't really know where to start.
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1answer
42 views

Adjoint operator between Hilbert spaces, does it map a subspace onto a subspace?

Let $X$ and $Y$ be Hilbert spaces and let $f\colon X \to Y$ be a linear continuous bijection. Define the adjoint $f'\colon Y \to X$ by $(f'y, x)_X = (y, fx)_Y$. Let $X_0$ and $Y_0$ be subspaces of ...
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2answers
74 views

Completeness of the space of random variables with bounded conditional first moment with respect t0 $\left\Vert \cdot\right\Vert _{2} $

Consider a probability space $\left(\Omega,\mathcal{F},P\right) $, and a sub-sigma-algebra $\mathcal{G}\subseteq\mathcal{F} $. As usual, let $L^{2}\left(\Omega,\mathcal{F},P\right) $ be the space ...
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53 views

Eigenvector of a linear combination of operators is an eigenvector of each operator

Assume $H$ is a Hilbert space and $a_1,\dots,a_n$ are operators with Hermitian adjoints $a_1^*,\dots,a_n^*$, satisfying the canonical commutation relations. Define $N_j=a_j^*a_j$. Assume $v$ is an ...
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1answer
166 views

Visualize and define a vector space without dot / inner product

I'm trying to rebase my know how in linear algebra, restart from scratch to get a more formal and useful set of definitions to help me with computer programming stuff . One of the first concepts is a ...
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69 views

Selfadjointness of the differential operator in a singular potential

The free Dirac operator is the differential operator of the following form $$ T_0 = i \alpha \nabla + \beta,$$ where $\alpha$ and $\beta$ are Hermitian $4 \times 4$ matrices, and $T_0$ is selfadjoint ...
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1answer
61 views

Riesz map on $L^2(0,T;H)$ — it's not “unique” in a way

Let $R:L^2(0,T;H) \to L^2(0,T;H^*)$ denote the Riesz representation map. Given $u \in L^2(0,T;H)$, $Ru \in L^2(0,T;H^*)$ can be changed in $[0,T]$ on a set of measure zero. So $Ru$ is not unique in ...
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88 views

Modes of convergence in infinite direct sums of $L^{2}$ spaces

It is known that if a sequence of random variables converges in norm then there exists a subsequence which converges almost surely. That is: let $\left(X_{n}\right)_{n\in\mathbb{N}}\subseteq ...
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1answer
168 views

Proving an orthonormal set is an orthonormal basis in Hilbert space [duplicate]

Consider a separable Hilbert space $H$, and $\{g_n\}$ is an orthonormal basis of $H$. Now there is an orthonormal set $\{f_n\}$ that satisfies $\sum_n\|f_n-g_n\|^2<1$. Show that $\{f_n\}$ is also ...
4
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1answer
145 views

T is not compact operator

I want to show that if $T$ is a bounded operator between two Hilbert spaces and $T$ is not compact then there exists an orthonormal sequence $y_{n}$ and an $R>0$ such that $\forall n\in ...
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93 views

Equivalent definitions of the trace of a Hilbert-Schmidt operator

I am currently reading the book Spectral Methods in Automorphic Forms, and Iwaniec defines the trace operator in a different way than I am accustomed to. Throughout, assume that everything converges ...
5
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1answer
186 views

Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
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2answers
165 views

Spectral Measures: Spectral Spaces (I)

Problem Given a Hilbert space $\mathcal{H}$. Let the Lebesgue measure be $\lambda$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its ...
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1answer
98 views

If $z$ is in an inner product space $X$, show that $f(x)=\langle x,z \rangle$ defines a bounded linear functional $f$ on $X$.

If $z$ is any fixed element of an inner product space $X$, show that $f(x)=\langle x,z \rangle$ defines a bounded linear functional $f$ on $X$, of norm $||z||$. If the mapping $X\to X'$ given $z\to f$ ...
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96 views

Closure in a Hilbertspace

Define for a self-adjoint pure contraction $S$ (remember: $\|S\|\leq1$ and $\pm1\notin\sigma_p(S))$ on a Hilbert space $\mathcal{H}$ the following set: $C_c^*(S):=\{g(S):g\in C_c(\hat{\sigma}(S))\}$ ...
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1answer
65 views

Dense Subspace: ONB

This might be a duplicate. If so, then please let me know. Thanks! Given a Hilbert space $\mathcal{H}$. Consider a dense subspace $\overline{Z}=\mathcal{H}$. Then it provides an ONB: ...
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96 views

Spectral Measures: Riemann-Lebesgue

Problem Given a Hilbert space the Lebesgue measure. Consider a selfadjoint Hamiltonian: $$H:\mathcal{D}\to\mathcal{H}$$ Denote its associated Borel spectral measure by: ...
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1answer
56 views

Reducing Spaces: Complement

Given a Hilbert space $\mathcal{H}$. Consider an operator: $$T:\mathcal{D}(T)\to\mathcal{H}:\quad\overline{\mathcal{D}(T)}=\mathcal{H}$$ Regard a subspace: ...
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1answer
19 views

Ordered Projections: Range

Given a Hilbert space $\mathcal{H}$. Consider two orthogonal projections $P,Q$. Then: $$P\leq Q\implies\mathcal{R}(P)\subseteq\mathcal{R}(Q)$$ The ordering being induced by: ...
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2answers
112 views

a question about contractions on Hilbert spaces

Let $\cal{H}$ be a Hilbert space, $T_1,T_2\in\cal{B(H)}$, $\|T_1(h_1)+T_2(h_2)\|^2\leq\|h_1\|^2+\|h_2\|^2$ for all $h_1,h_2\in\cal{H}$. $T_1T^\ast_1+T_2T^\ast_2\leq I$. Then can we verify that 1 ...
3
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1answer
65 views

Normal Operator: Everywhere defined implies bounded?

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{H}\to\mathcal{H}$. If its domain is the whole Hilbert space then is it necessarily bounded? The point is that I'm trying ...
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2answers
194 views

Is $\|x\| = \| \overline{x} \|$ in an inner product space?

Suppose $X$ is a complex inner product space of complex valued functions that is closed under conjugation. Is it true that $\|x\| = \| \overline{x} \|$ for all $x$? If not, is there a simple ...
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1answer
44 views

The projection onto a subspace of a Hilbert space is the nearest point

Let $H$ be a Hilbert space, $M$ its linear subspace, and $p_M$ the projection onto $M$. Prove the following: for every $v \in H$ and every $u \in M$, $$\|v-p_Mv \| \leq \|v-u\|$$ with equality only ...
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1answer
43 views

Isometric Operators: Common Core

Given a Hilbert or Banach space $\mathcal{H}$. Consider two closed operators $S:\mathcal{D}(S)\to\mathcal{H}$ and $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose they're isometric on a common core ...
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1answer
105 views

$\langle Tx,x\rangle =0$ , then T is zero

I just wanted to be sure about something. The implication $\langle Tx,x\rangle =0$ , then T is zero , holds only if $T$ is self-adjoint right? If $T$ is an arbitrary operator, we need to have $\langle ...
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1answer
30 views

Bounding a linear functional in $L_2[0, 1]$

For each f in $L_2[0, 1]$ let $\phi(t)$ be the solution of $y' + ay = f$ that satisfies $\phi(0) = 0$, where a is a constant. Define $l: L_2[0,1] \to \mathbb{C}$ by $l(f) = \int_0^1 \phi(t) dt.$ ...
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1answer
103 views

Gelfand triple: what happens if we don't identify the pivot Hilbert space with its dual?

People usually say $V \subset H = H^* \subset V^*$ is a Gelfand triple if $V$ is continuously and densely embedded in $H$ and $H$ is identified with its dual. Sometimes they do not mentioned that ...
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0answers
98 views

Making a complex inner product symmetric

Let $(V, (\cdot, \cdot))$ be a complex inner product space, say a space of complex-valued functions, with $(\cdot, \cdot)$ linear in the second position and sesquilinear in the first. Assume that $V$ ...
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1answer
34 views

Evaluating the $L_2[-1, 1]$ inner product on rescaled Legendre polynomials

Let $z_n(t) = \sqrt{\frac{2n+1}{2}} \frac{1}{2^n n!} \frac {d^n}{dt^n} (t^2-1)^n$, a rescaled Legendre polynomial. As an intermediate step of a larger problem, I need to show that in terms of the ...
3
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3answers
172 views

Selfadjointness of the Dirac operator on the infinite-dimensional Hilbert space

I am a physicist, so my background in functional analysis is limited only to basics. However, I would like to prove that the free Dirac operator is selfadjoint (or Hermitian, or neither). The free ...
3
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1answer
35 views

Family of sequences in a Hilbert space with certain property

Suppose $\mathcal{F}$ is a family of sequences on the unit sphere of $l_2$ with the following property: For any sequence $\varepsilon_n\downarrow 0$ but which is not eventually identically $0$, there ...
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1answer
23 views

on Limits and sequences proofs of a closed subspace of the hilbert space.

$M$ is a closed subspace of the Hilbert space $H$ and $ x\in H$ My book states these two claims. (1) If $d = \inf_{y \in M} \|x - y \|^2 $then there is a sequence of elements $\{y_n\}$ of $M$ such ...
3
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2answers
49 views

Finding the maximum of an integral of a function with given constraints.

This comes from Rudin's Real Analysis text. The first part of the problem asks us to compute $\displaystyle\min_{a,b,c}\int_{-1}^1|x^3-a-bx-cx^2|dx$ (which I have done). Now it asks us to find ...
2
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3answers
453 views

Hermitian and self-adjoint operators on infinite-dimensional Hilbert spaces

I am a physicist and I am trying to get a grasp on the following terms from functional analysis: As I understand, an operator is Hermitian if it is symmetric and bounded (domains of A and A* don't ...