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

learn more… | top users | synonyms

5
votes
1answer
191 views

Conditions for the sequence being weakly convergent

Let $H=\ell_2$ be the Hilbert space of the square-summable sequences where $$ \langle x,y\rangle=\sum_{i=1}^{\infty}x_iy_i, \quad \|x\|=\sqrt{\langle x,x\rangle}. $$ Let $F: H\rightarrow H$ be an ...
1
vote
2answers
70 views

How do functions form a Hilbert space

I'm trying to wrap my head around function spaces. I get that you can define the inner product as the integral the multiplication of two functions over the entire domain because it satisfies the ...
1
vote
2answers
48 views

Why is the sequence $(\langle x_n,a \rangle)$ Cauchy when $(x_n)$ is?

Let $\mathcal H$ a Hilbert space over $\mathbb R$ and $A = \{x\in \mathcal H : \langle x, a \rangle \geq 1 \}$. I'm trying to prove that $A$ is closed. Let $(x_n) \subset A$ be a Cauchy-sequence. ...
0
votes
1answer
103 views

About the positivity of the inner product on $L^2[0,1]$

My textbook on Hilbert space theory claims that the map $$\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}~\mathrm{d}x$$ is an inner product on $C[0,1]$. But I am not sure whether ...
1
vote
1answer
224 views

Unique extension to a bounded operator

Suppose $\left\{ e_{1},e_{2},\ldots\right\} $ is an orthonormal basis for a Hilbert space $\mathcal{H}$ and for each $n$ there is a vector $Ae_{n}$ in $\mathcal{H}$ such that $\sum\left\Vert ...
3
votes
1answer
212 views

On the regularity of the Laplace equations and tensor products and such

To start with, let me apologize for my ignorance as I know next to nothing about partial differential equations. My question is about the tensor product of Banach spaces but actually I do not ...
4
votes
1answer
122 views

Showing a representation is irreducible by showing that a degenerate subspace has codimension one.

Throughout $\phi$ be a continuous character from a locally compact abelian group $G$ to the circle. I'm trying to understand this implication. Basically we want to show that a certain representation ...
2
votes
2answers
124 views

Theorem about orthogonal system in inner product space.

It is known that "If $\{x_n\}$ is a sequence in a real Hilbert space $H$ satisfying $$ \langle x_n, x_m\rangle =0 \quad\forall n\ne m, $$ then $\displaystyle\sum_{n=1}^{\infty}x_n$ is convergent if ...
2
votes
1answer
131 views

Finding a linear mapping in a special Hilbert space

Let $H=\ell_2$, the real Hilbert space whose elements are the square-summable sequences of real scalars, i.e., $$ H=\left\{u=(u_1,u_2,\ldots,u_i,\ldots): ...
0
votes
0answers
55 views

On Scalar products — What is wrong with this argument?

Suppose $|k\rangle =\exp(k \hat O^\dagger)|0\rangle$ where $c_k\in \mathbb C$ and $|0\rangle$ is normalized. I wish to evaluate $\langle a|b\rangle$. Here is what I think, but the result is not ...
1
vote
1answer
181 views

Countable Hilbert Spaces

I have seen a simple proof that no banach space over $\mathbb{R}$ can be of countably infinite dimension. However since the space of all square integrable functions on the unit interval forms a ...
3
votes
1answer
141 views

What is meant by `element $x\in H$ of minimal norm'

I do not seek a proof of the following exercise. I just want to understand this question in order to solve it myself. Let $H$ be a Hilbert space over $\mathbb R$ and let $a, b\in H$ be such that ...
2
votes
0answers
130 views

Convergence of orthogonal projections

Suppose that $a_m$, $m \in \mathbb{N}$, is a sequence of bounded linear operators on a Hilbert space converging strongly to an bounded linear operator $a$. If U is a finite-dimensional subspace of H, ...
3
votes
6answers
184 views

Is $\operatorname{range} =\ker^\perp$ only true for projection?

Let $P$ be a linear operator on a Hilbert space $H$. If $\operatorname{range} P=(\ker P)^\perp$, is $P$ necessarily a projection, i.e., $P^2=P$?
3
votes
1answer
518 views

The relation between bounded invertible and surjective operators

Please, answer me that how is the set of all bounded invertible operators (for example on a Hilbert space) clopen (closed and open) in the set of all bounded surjective operators? In fact, which ...
2
votes
2answers
138 views

$f'$ is in $L^2[0,1]$

Let $f$ is absolutely continuous function on $[0,1]$, $f(0)=0$ and $f' \in L^2[0,1]$. Would you help me to prove that there is constant $c$ such that $$|f(t)| \leq c \left( \int_0^1 |f'(t)|^2 dt ...
1
vote
1answer
69 views

Convergent series with coefficient in $\ell^2$.

Let $z$ denote a complex number and $\{\alpha_n\}$ be a sequence in $\ell^2$. Would you help me to prove that series $\sum_{n=0}^{\infty} \alpha_n z^n$ has radius of convergence greater than or equal ...
4
votes
1answer
414 views

Spectral theorem for unitary operators

I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
6
votes
3answers
298 views

A complete orthonormal system contained in a dense sub-space.

Let H be a separable complex Hilbert space. Let A be a dense sub-space of H. Is it possible to find a complete orthonormal system for H that is contained in A?
1
vote
1answer
108 views

Contractibility of the sphere and Stiefel manifolds of a separable Hilbert space

Why are the sphere $$S=\lbrace |x|=1\rbrace$$ and the Stieffel manifolds of orthonormal $n$-frames $$V_n=\lbrace (x_1,\dots,x_n)\in S^n\mid i\neq j\Rightarrow\langle x_i|x_j\rangle=0\rbrace$$ of a ...
7
votes
2answers
2k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
4
votes
2answers
147 views

Orthogonality checking in Kreyszig exercise

Let $H$ be inner product space with inner product $\langle\cdot,\cdot\rangle$ and norm $\lVert \cdot\rVert$. Let $x,y \in H$. Would you help me to prove that $\langle x,y\rangle=0$ if and only if ...
0
votes
1answer
340 views

is a direct sum of Hilbert spaces a Hilbert space.?

I have read in proofwiki that a direct sum of Hilbert spaces is a Hilbert space. However, Wikipedia Page about direct sum says it is not necessarily true, that is, the direct sum of Hilbert spaces is ...
0
votes
0answers
116 views

Set of Bounded linear Operators on $l_2$ is dense on the set of bounded operators on $l_2$?

Let $l_2^{+}$ be the Hilbert space of all square summable sequences $\{x_n\}, n \in \mathbb{N}$ under some definition of inner-product $\langle,\rangle_l$. Define $B[l_2^{+}]$ as the set of all ...
1
vote
1answer
204 views

Proving Fréchet differentiability

Am learning about Fréchet differentials and was wondering if for a real matrix $X$ and positive semidefinite real matrices $A,B$ the function $f(X)=TrX^TAX-X^TBX$ is twice Fréchet differentiable or ...
0
votes
1answer
184 views

Solution for a Frobenius norm inequality

Am trying to find a real scalar $\gamma$ such that for a given pair of real rectangular matrices $X,Y$ the following holds: $\frac{||Y||_{F}^{2}}{5} \leq ||\gamma X||_{F}^{2}\leq ||Y||_{F}^{2}$ ...
0
votes
1answer
78 views

Scalar multiplication and Frobenius norm

Was wondering on what would be the real number (scalar) $\gamma$ that needs to be multiplied with each entry in a real rectangular matrix $X_{m\times n}$ such that the Frobenius norm of $X$ equals a ...
4
votes
1answer
83 views

Invertible operator not preserving Hilbert dimension

It is known that for a bijective linear operator $T:X\to Y$ the algebraic dimensions of the linear spaces $X$ and $Y$ coincide. I am asking for an example of an invertible (bounded) linear operator ...
1
vote
0answers
464 views

Proof of Isometry: Inner Product Preserving Map

For known points $x_i,x_j,\ldots,x_k$, in $\mathbb{R}^n$, consider a mapping $y_i,y_j,\ldots,y_k$ in $\mathbb{R}^n$ produced by minimizing the function $f(y)=\sum_{i,j} \left \langle x_i,x_j \right ...
2
votes
2answers
76 views

Fourier Coefficients in arbitrary Hilbert Spaces

Say we have an orthonormal basis $\{e_n\}$ for a infinite Hilbert Space $H$. I want to prove that any vector $x=\sum_{n=1}^\infty\langle x, e_n\rangle e_n$. I don't know where to start. Could I ...
2
votes
1answer
90 views

Can this Lemma be extended a little?

Consider this lemma (my question are below): Lemma Given three pairwise orthogonal subspaces $X$, $Y$, $Z$ of a Hilbert space $H$ that span the whole space, any vector $\nu\in H,\ ||\nu||=1$, can ...
4
votes
1answer
650 views

Formula for trace of compact operators on $L^2(\mathbb{R})$ given by integral kernels?

Given an appropriate function $K: \mathbb{R}^2 \to \mathbb{C}$, say continuous of compact support, we obtain a compact operator $T$ on the Hilbert space $L^2(\mathbb{R})$ by the formula $$ (T h)(t) = ...
2
votes
0answers
143 views

A doubt about tensor product on Hilbert Spaces

An operator is a bounded (i.e., continuous) linear transformation between Hilbert spaces. Let $\mathcal{B}[\mathcal{H}]$ be the set of all operators in the Hilbert space $\mathcal{H}$. Let ...
2
votes
1answer
230 views

orthogonal subspaces in a Hilbert space

Is it true that if $A,B$ are closed subsets of a Hilbert space $H$, such that $A\perp B$, we have $A+B+(A\cup B)^{\perp} =H$ ? What if $A,B$ are closed subspaces ?$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ...
0
votes
1answer
71 views

Rearragement of a series in Hilbert space

Let $H$ be a Hilbert space and $\sum_k x_k$ a convergent infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation ...
1
vote
1answer
47 views

Can this type of series retain the same value?

Let $H$ be a Hilbert space and $\sum_k x_k$ a countable infinite sum in it. Lets say we partition the sequence $(x_k)_k$ in a sequence of blocks of finite length and change the order of summation ...
5
votes
2answers
864 views

Relationship of Fourier series and Hilbert spaces?

I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...
0
votes
1answer
102 views

is this a subset of the set of all selfadjoint operators?

Is the set $Q$ of all operators that have the property that the image is orthogonal to the kernel, and the kernel isn't the null space, only a subset of the set $T$ of the selfadjoint operators or ...
2
votes
0answers
99 views

RKHS concepts. Connection with matrix trace function.

With my basic linear algebraic background, am trying to connect the Reproducing Kernel Hilbert Space (RKHS) concepts to example functions over matrices, as I gradually learn about this new concept. ...
1
vote
1answer
135 views

Reproducing Kernel Hilbert Space- notation and basics

Am reading about Reproducing Kernel Hilbert Space(RKHS) while reading through Functional Analysis and Hilbert Space material and am unable to get the notation : $k(·,xi)$ correctly. What does the dot ...
0
votes
0answers
119 views

Is the closure of a symmetric operator unique?

Let $T:D(T)\rightarrow H$ be a densely defined symmetric operator in a Hilbert space H. The closure $\overline T$ of $T$ is defined as the operator whose graph $G(T)$ is the closure of the graph of ...
1
vote
1answer
180 views

Representing with Hilbert Schmidt Norm

Am trying to see, if the following Trace function can be expressed using a Hilbert Schmidt Norm: $\operatorname{Tr}(X^TAX)$. Here, $X$ is a matrix whose entries take values that are finite and reals ...
2
votes
3answers
552 views

Compact operators and uniform convergence

Suppose $T: H \rightarrow H$ is a compact operator, $H$ is a Hilbert space, and let $(A_n)$ be a sequence of bounded linear operators on $H$ converging strongly to $A$. Show that $A_nT$ converges in ...
16
votes
2answers
4k views

Finding the adjoint of an operator

This is from my homework, I'm totally lost as to how to proceed. Consider the operator $T: L^2([0,1]) \rightarrow L^2([0,1])$ defined by $(Tf)(x) = \int^x_0 f(s) \ ds$ What is the adjoint of $T$? ...
2
votes
1answer
224 views

Range of identity plus compact operator is closed

Suppose $K:H\to H$ is a compact linear operator on a Hilbert space $H$. How do I show that the range of $I+K$ is closed in $H$? I believe this is equivalent to showing that $\{x_n\}\subset H$ and ...
3
votes
1answer
169 views

Operator norm of the sum of a finite collection of bounded linear operator

I recently got some difficulty with my homework question. The question is: Let $T_1,\dots,T_N$ be a finite collection of bounded linear operators on a hilbert space $H$, each of operator norm $\le ...
1
vote
1answer
198 views

Direct sum of compact operators

I am having some trouble proving this: Let $T_1\in H_1$ and $T_2\in H_2$ where $H_1,H_2$ are Hilbert spaces. Let $T=T_1\oplus T_2$ on $H_1\oplus H_2$. I need to show $T$ is compact iff $T_1$ and $T_2$ ...
3
votes
2answers
514 views

Hellinger-Toeplitz theorem use principle of uniform boundedness

Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...
1
vote
0answers
42 views

Radial and angular part of the space of compactly supported smooth functions

On page 124 of Thaller's The Dirac equation the following space is mentioned : $$C^{\infty}_0(0, \infty)\otimes C^\infty(\mathbb{S}^2)\subset L^2(0, \infty)\otimes L^2(\mathbb{S}^2),$$ where the ...
4
votes
1answer
117 views

Is it possible to 'approximate' compact, convex sets in $\ell^2$ by the Hilbert cube

Define $H=\{(x_n)_n\in\ell^2:|x_n|\le \frac1n, n\in\mathbf N\}\subset\ell^2$. This set is known as the Hilbert cube and it is well-known that $H$ is compact, convex and non-empty. Let ...