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

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57 views

Alternative explanation for $\iint_D \left|\log \left( \frac{e}{1-z} \right) \right|^2 \ dA = \frac{\pi^3}{6}$?

I thought up a curious definite integral. Let $D = \{ z \in \mathbb{C} : |z|<1\}$. Let $A$ denote area measure on $D$, normalized so that $A(D) = \pi$. I claim that $$\iint_D \left|\log \left( ...
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0answers
73 views

If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
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1answer
536 views

Relation between adjoint operators/dual operators

I'm a bit confused about adjoint operators. Let $T:X \to Y$ be a linear isomorphism between Hilbert spaces. Then is it true that $(Tx,y)_Y = (x,T^*y)$ exists (does $T^*:Y \to X$ always exist)? What ...
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1answer
108 views

Connecting two Hilbert spaces' inner products via isomorphism

If I have two Hilbert spaces $X$ and $Y$ and a continuous linear isomorphism $T:X \to Y$ with continuous inverse $T^{-1}:Y\to X$, is there anyway to write $$(a,b)_X$$ as an inner product on $Y$? I ...
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0answers
85 views

Adjoint operator on subspace

Let $T:V_1 \to V_2$ be linear with adjoint $T^*:V_2^* \to V_1^*$. Suppose $V_i \subset H_i \subset V_i^*$ is a Hilbert triple. Let $f \in H_2 \subset V_2^*$. How can I interpret $T^*f$? Is it just ...
0
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1answer
148 views

Hilbert norm and Euclidean distance

For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
5
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1answer
171 views

Convex subset of Hilbert space as intersection of closed balls

How does one prove that any closed, convex, and bounded subset of a Hilbert space is the intersection of the closed balls that contain it?
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2answers
140 views

Linear operator and extension of its inverse

Let $K:H_1 \to H_2$ be a linear operator between Hilbert spaces that may not be bounded. $K$ is bounded below. So $K$ has an inverse $K^{-1}:\text{Range}(K) \to H_1$. $K^{-1}$ extends by ...
3
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1answer
37 views

Essential selfadjointness preserved under unitarily transfomration?

I am wondering if essential selfadjointness of an operator in a Hilbert space is preserved under unitarily transformations. In other terms: let $H,H'$ be two isomorphic Hilbert spaces, with an ...
2
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1answer
78 views

Hilbert space image of basis under bicontinuous map

Let $X$ and $Y$ be separable Hilbert spaces and $T:X \to Y$ be linear continuous with linear continuous inverse $T^{-1}:Y \to X$. If $x_n$ is a countable orthnormal basis of $X$, then can I say that ...
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1answer
670 views

Density and closedness of $C[0,1]$ in $L^\infty[0,1]$ in norm and weak-* topologies

With results: "For convex subsets of a locally convex space, a, originally( strongly) closed equals weakly closed, and b, originally (strongly dense equals weakly dense." Could you help me solve this ...
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0answers
47 views

About the problem 20 chap 3 (functional analysis, Walter Rudin) [duplicate]

Possible Duplicate: I need to show that $K$ is compact and that $co(K)$ is bounded, but not closed. Let $\{u_1,u_2,u_3,\dots \}$ be sequence of pairwise orthogonal unit vectors in Hilbert ...
2
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1answer
140 views

The closed unit disk in an infinite dimensional Hilbert space has a closed subspace homeomorphic to $\mathbb R$

Let $V$ be a Hilbert space, $D^{\infty}$ is the closed unit disk in an infinite dimensional Hilbert space $V$. Prove that $D^{\infty}$ has a closed subspace homeomorphic to $\mathbb{R}$. I've found ...
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1answer
110 views

Picard Condition (searching for an idea)

The so-called Picard-condition is: Let X,Y be Hilbertspaces and $T\colon X\to Y$ is a compact operator with singular value decomposition system $\left\{(\sigma_j,u_j,v_j)\right\}$. An element ...
4
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1answer
73 views

Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?

Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
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0answers
38 views

A question about a relationship of expressions got from change of variables/inner products

Suppose $F:L^2(S) \to L^2(T)$ is linear homeomorphism such that $F(v) = v \circ \mathcal{F}$ where $\mathcal{F}:T \to S$ is a diffeomorphism. Suppose $$\lVert F(v) \rVert_{L^2(T)} \leq C\lVert v ...
1
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1answer
618 views

Isometric isomorphism of Hilbert spaces and orthonormal basis

If I have an isomorphism of two separable Hilbert spaces that preserves norms, does the isomorphism map orthnormal basis to orthonormal basis? I can't show it.
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1answer
45 views

‎‎$‎\langle ‎(‎x_{n}‎)‎,(y_{n})\rangle=\sum_{‎1‎}^{‎\infty‎}\frac{‎‎x_{‎n‎}‎‎\bar{y_{‎n‎}}}{n^{2}}‎$‎‎ defines an inner product

Check ‎that ‎the ‎formula ‎‎$‎\langle ‎(‎x_{n}‎)‎,(y_{n})\rangle=\sum_{‎1‎}^{‎\infty‎}\frac{‎‎x_{‎n‎}‎‎\bar{y_{‎n‎}}}{n^{2}}‎$‎‎ defines an inner product ‎on ‎‎$‎\ell‎^{‎\infty‎}‎$‎,‎ ‎the space of ...
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1answer
84 views

Parseval type identity

I have an orthonormal system of functions $$ U = \left\{ u_{\lambda}(x) \in L_{2}(\mathbb{R}_{+}) \mid \lambda \in \left\{-1,\ldots,-n\right\} \cup\mathbb{R}_{+} \right\} $$ such that for any $f,g ...
2
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1answer
414 views

A generalization of the Cauchy-Schwarz inequality to linear operators

If $A$ is an operator and $A \in \mathcal{B_{+}(X)}$ (the set of the positive operators) then the generalization of the Cauchy-Buniakowsky-Schwarz inequality holds: $$|\langle Ax,y\rangle| \leq ...
3
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1answer
491 views

about closed linear subspace

Can you help me, plese, with the notion of closed linear subspace. What means, examples of closed linear subspace, how can I prove that a subspace is a closed linear subspace. Thanks :-)
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0answers
63 views

A set of trajectories as a linear subspace of Hilbert space

Let $\left\{S(t)\right\}_{0 \leqslant t \leqslant \theta}$ be a strongly continuous semigroup of linear continuous operators in Hilbert space $H$, $S(0) = I$. Let $x$ be some element of $H$. Then its ...
3
votes
2answers
287 views

Find adjoint operator of an operator T

I would like to find the adjoint operator of $$ T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds. $$ Here $H^1([0,1])$ is the Sobolev space $W^{1,2}([0,1])$. I tried to find ...
3
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1answer
1k views

Orthogonal projection on the Hilbert space .

I want to prove the following: If $X$ is a Hilbert space and $Y$ is a closed subspace of $X$, then every $x\in X$ can be written as $x=y+z $ where $y\in Y$, $z \in Y^\perp$. The ...
4
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1answer
761 views

How to find an orthonormal basis for $L^2(\mathbb{R},\mathbb{C})$?

Consider the Hilbert space $X:=L^2(\mathbb{R},\mathbb{C})$ Now consider the operator that takes the second derivative, i.e. $A := \partial_{x}^2$, i.e. $A: H^2(\mathbb{R},\mathbb{C}) ...
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0answers
36 views

Lie Derivative in Projective Hilbert Space

In considering a projective Hilbert space, $P(H)$, for linear maps (tensors) of vectors in the space, $A^{a}_{b}v_{a}=u_b$, is there a natural definition for the Lie Derivative for such linear maps? ...
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2answers
947 views

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 ...
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0answers
134 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 ...
2
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1answer
397 views

Conditions on the weight function for Hermite polynomials' completeness

Hermite polynomials form a complete orthonormal basis of the weighted $L^2(\mathbb R, w \; dx)$ space, with inner product $$ \langle f, g \rangle_w = \int_\mathbb R f(x) g(x) \; w(x) dx. $$ A short ...
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3answers
206 views

Functional analysis-Hilbert spaces

Let $ X$ be an inner product space. Show that $ X$ is a Hilbert space if and only if for each continuous linear functional $ L$ on $ X$,there exists $ z\in X$ such that $ L(x)=\langle x,z\rangle $ . ...
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1answer
147 views

Functional analysis - bounded linear transformation

Let $ \mathcal{H} $ be a Hilbert space, and let $ T: \mathcal{H} \to \mathcal{H} $ be such that $ \langle x,Ty \rangle = \langle Tx,y \rangle $ for all $ x,y \in \mathcal{H} $. How can one show that ...
3
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1answer
226 views

Bounded operators on separable Hilbert spaces

Let $H$ be a separable Hilbert space. Show that every bounded operator from $H$ to itself can be approximated in the strong operator topology by a sequence of finite rank operators. Im not sure what ...
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1answer
397 views

Trace class for operators

Let $ \mathcal{H} $ be a Hilbert space and $ T: \mathcal{H} \to \mathcal{H} $ a bounded linear operator. The $ n $-th singular number $ {\mu_{n}}(T) $ of $ T $ is defined as the distance from $ T $ ...
5
votes
2answers
607 views

Norm inequality for sum and difference of positive-definite matrices

If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear ...
4
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2answers
1k views

What is the difference between a complete orthonormal set and an orthonormal basis in a Hilbert space

I don't know if it's true but I think in a finite dimensional veccter space, an orthonormal set which is complete becomes a basis. But in a Hilbert space, the books say that the set of finite linear ...
4
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2answers
96 views

A counterexample on the existence of some sequence in Hilbert space

I want to find a uniformly bounded sequence $\{x_n\}$ in $l^2(\mathbb{C})$ such that $x_n$ does not converge to zero in weak topology, i.e., $\exists ~y\in l^2(\mathbb{C}),$ such that $\langle y, ...
2
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1answer
111 views

Normal Operator surjective

I have difficulty proving: If $T$ is a normal operator in a Hilbert space, $T$ is surjective if and only if $T^*$ surjective. Please give me some help. Thank you.
4
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1answer
168 views

Multiplication operator and trace class

Suppose we work in $H=l^2(\Bbb{N})$ and suppose the multiplication operator $T_f$ such that $T_f\psi=f\psi$ and $f:\Bbb{N}\rightarrow \Bbb{C}$. We denote by $B_1(H)$ the trace class of operators. ...
4
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1answer
223 views

Proof that the spectrum of the Dirichlet Laplacian is discrete

Let $\Omega\subset\mathbb{R}^n$ a open bounded set. The Dirichlet laplacian can be defined via it's closed semi-bounded form on $H^1_0(\Omega)$. The fact that it's spectrum is discrete is as far as I ...
5
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2answers
283 views

Brownian Motion Covariance: max instead of min

It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
4
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0answers
157 views

What is the dual space of $C([0,T];X)$ ($X$ Hilbert space)?

What is the dual space of $C([0,T];X)$, where $X$ is a Hilbert space? Is it $\operatorname{BV}([0,T]; X^*)$? As we know, for $C([0,T])$, the dual space is $\operatorname{BV}([0,T])$, but when it is ...
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2answers
312 views

Does a square-integrable function always have the derivative of its integral over one variable equal to zero?

In quantum mechanics, one requires that $$\frac{d}{dt}\int_{-\infty}^\infty\left|\psi(x,t)\right|^2dx=0$$ in order for normalization to be independent of time. In general, is it true that for any ...
4
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1answer
230 views

Spectrum in an separable Hilbert space

Let $H$ be a separable Hilbert space with orthonormal basis $\{e_i\}$. Let $(c_n)$ be a bounded sequence of complex numbers and consider the bounded linear operator $T$ on $H$ defined by $$Tx = ...
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1answer
316 views

Hilbert space of absolutely continuous functions

Let $H$ be the space of functions $\alpha: [0, T] \longrightarrow \mathbb{R}^n$ that are absolutely continuous and such that $\alpha(0)=0$. The statement that I have implicitly found in a paper is ...
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1answer
537 views

Minimizing sequence

This came up in a proof I was reading. Define $$\inf_{z \in K} \|x-z\| = d$$ Let $y_n\in K$ be a minimizing sequence How do we know that such a minimizing sequence exists? Here K is a closed convex ...
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1answer
98 views

Differential Operator on $L_{2}$ problem

I am working on a problem from a textbook and have run into difficulties on this specific question. Any assistance will be appreciated, Consider the partial differential equation, $\frac{\partial ...
1
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0answers
85 views

Extension of differentiation operator to $L_2[0,1]$.

I'm studying for my functional analysis exam. We are required to know the proof of the following, but I cannot figure it out. Consider $L_2[0,1]$ with orthonormal basis $(e_n)_{n=-\infty}^\infty$ ...
4
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1answer
93 views

Is this function positive?

I was wondering if: $$\int_0^1x(t)\int_0^tx(s)ds\ dt$$ is positive for a general $x\in L_2[0,1]$ . Can you help me with this?
2
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1answer
65 views

Finding the minimizing vector of a $l_{2}$ sequence

I am working on a problem sheet and this question has me stuck. A little guidance will be appreciated. Let $X = l_{2}$. Let $x \in X$ be given by $x = \{\frac{1}{2^{i}} \}^{\infty}_{i=1}$ Let $M ...
1
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1answer
156 views

What is this Hilbert space?

The space is $H^s(\mathbb R^d)$. If $f$ is in this space, it means $\int_\mathbb {R^n} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi < \infty$ where $\hat f$ is the fourier transform of $f$: $\hat ...