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

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3
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1answer
82 views

How to show projection of $L^2$ function converges to that $L^2$ function

My teacher said that if $P_n f = \sum_{j=0}^n(f,w_j)w_j$, where $w_j$ is orthonormal basis of $L^2$, then $|P_n f- f|_{L^2} \to 0$ for $f \in L^2$. How do I prove this? I thought $$|P_nf - f| = ...
0
votes
1answer
109 views

Is an orthnormal basis of $L^2([0,1])$ also an orthonormal basis of $L^2((0,1))$?

My question is: If $\lbrace e_n \rbrace$ is an orthnormal basis of $L^2([0,1])$, is $\lbrace {e_n}_{|(0,1)} \rbrace$ an orthonormal basis of $L^2((0,1))$? As the points $\lbrace 1 \rbrace$ and ...
2
votes
2answers
137 views

Uncountable union of separable spaces is separable?

If $V(x)$ is a separable Hilbert space, is $\bigcup_{x \in X}V(x)\times\{x\}$ separable when $X$ is an uncountable set? How to make it separable if it's not? What assumptions do I need?
4
votes
1answer
60 views

Selfadjoint and continuous operator on a complex Hilbert space

Let $T\colon H\to H$ be a selfadjoint continuous operator on a complex Hilbert space. Show: $$ \lVert (T\pm i\mbox{Id})x\rVert^2=\lVert Tx\rVert^2+\lVert x\rVert^2~\forall~x\in H. $$ -- How can I ...
2
votes
2answers
532 views

Isomorphism of Banach space

If $T:H\to B$ is isomorphism of Banach spaces and $H$ is Hilbert, must $B$ necessarily be Hilbert?
2
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0answers
143 views

Find the adjoint operator

I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator $$ (Ax)(t)=x(at), x\in L^2(0,\infty), a>0. $$ My calculation is the following; I use the ...
5
votes
2answers
850 views

Parallelogram law valid in banach spaces?

It is known that the parallelogram law $\|x-y\|^2+\|x+y\|^2 = 2(\|x\|^2 + \|y\|^2)$ holds in any space with an inner product (the norm being induced by this inner product). Is this formula valid in ...
1
vote
2answers
330 views

Equivalence of norms implies isomorphism between Hilbert spaces

If I have 2 Hilbert spaces with 2 norms, and a map between the Hilbert spaces, and I know that the norms are equivalent, does this mean that the spaces are isomorphic?
1
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0answers
49 views

Hilbert basis of $L^2([-1,1])$?

Could you please specify hilbert basis of $L^2([-1,1])$? How will be the representation of a function f $\in L^2([-1,1])$ by means of its Fourier series? My solution: $E_k=1/\sqrt2 e^{kit\pi}, k\in ...
2
votes
1answer
45 views

$V \subset H \subset V^*$, what's $\langle h, v \rangle_{V^*, V}$?

If $V \subset H \subset V^*$ is Hilbert triple, and $h \in H$ what's $\langle h, v \rangle_{V^*, V}$? I know we interpret it to be $(h,v)_H$. But is this correct: $$\langle h, v \rangle_{V^*, V} := ...
1
vote
1answer
99 views

Closed subset of a Hilbert space

$Y_0\subset Y $ is a closed Hilbert subspace of $Y$ with finite codimension and a subspace $ Y_1 $ satisfies $ Y_0 \subset Y_1 \subset Y $. Is $ Y_1 $ also closed?
4
votes
2answers
269 views

Trying to understand Hilbert Spaces…

I am trying to get a hold on Hilbert Spaces, but I am having difficulties combinging various definitions. I have looked it up on wikipedia and wolfram, there it states something like "A Hilbert ...
0
votes
0answers
130 views

Construct an orthonormal basis

Consider vector space of functions continuous on $I \subseteq \mathbb R$ and scalar product is defined as $({\bf f}, {\bf g}) \equiv \int_{I} \rho(x) f(x)g(x) dx$. Let the generating function $G(z,x)$ ...
3
votes
1answer
427 views

Estimate on the norm of a self-adjoint operator

EDIT: thks to Martin's comment I realize the previous version was wrong. Here is the correct version of what I need to show: I am trying to show that if $A$ is a self - adjoint operator in a Hilbert ...
1
vote
1answer
147 views

proof of RKHS for a particular kernel is unique

Suppose that I have a kernel $K$. Then show that the RKHS $H_1$ and $H_2$ of $K$ are the same. So I need to prove the above statement. To begin with, as an exercise, I proved the reverse statement ...
1
vote
1answer
154 views

Matrix Trace representation?

For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
3
votes
2answers
82 views

Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective.

Given a map $F:X \to X$ where $X$ is a Hilbert space, $F$ satisfying $f(x):=x-F(x)$ is a compact map. $\lim_{\|x\|\to \infty} \frac{(F(x),x)}{\|x\|} = \infty$ I'm seeking to prove that $F$ is ...
2
votes
2answers
189 views

elliptic pdes and associated bilinear forms for Lax-Milgram

I have a simple question on elliptic pdes, actually I can not understand clearly from definitions. Thats why I want to try think on an example. Let us have an elliptic pde $$-A \Delta ...
1
vote
1answer
56 views

Kernel inclusion implies factorization

I have a question whether a certain fact is true for arbitrary operators on a Hilbert space. Namely, consider Hilbert spaces $H,K$, an operator $A\in B(H)$ and another $B\in B(H,K)$. Moreover, assume ...
0
votes
2answers
52 views

Hilbert Space - Question about norm

Let $H$ be a Hilbert space. Is it true that, if $\|x\|$ is less than or equal to $r$ and $\|y\|$ is strictly greater than $r$, then $\left\| x-\frac{ry}{\|y\|} \right\|$ is less than or equal to ...
2
votes
1answer
53 views

Functional to inner product in Hilbert triple

If $V \subset H \subset V^*$ is a Hilbert triple, and $f \in V^*$, I cannot represent $f(v) = (e,v)_V$ because we don't identify $V$ with $V^*$. But is it true that $f(v) = (e,v)_H$ for some $e$?
1
vote
1answer
185 views

Subspace of a Hilbert Space

I am sitting through a course on "Operators in Hilbert Spaces". The instructor has asked us to look at the following problem: Let $H$ be a hilbert space and $E \subset H$. E is called weakly bounded ...
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0answers
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( ...
1
vote
0answers
72 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 ...
1
vote
1answer
491 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 ...
0
votes
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 ...
0
votes
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
votes
1answer
145 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
votes
1answer
169 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?
2
votes
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
votes
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
votes
1answer
77 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 ...
0
votes
1answer
640 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 ...
0
votes
0answers
44 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
votes
1answer
136 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 ...
1
vote
1answer
107 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
votes
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
vote
1answer
603 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.
0
votes
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 ...
1
vote
1answer
82 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
votes
1answer
387 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
votes
1answer
466 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 :-)
3
votes
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
283 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
votes
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
votes
1answer
727 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}) ...
1
vote
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? ...
17
votes
2answers
914 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 ...
1
vote
0answers
130 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 ...