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

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2
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2answers
131 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
133 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): ...
1
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1answer
216 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
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1answer
182 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
163 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
191 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
649 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
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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 ...
5
votes
1answer
509 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
413 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
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1answer
127 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 ...
8
votes
2answers
3k 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
178 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
399 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
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0answers
118 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
236 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
200 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
92 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 ...
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0answers
596 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
78 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
92 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
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1answer
875 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
157 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
249 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
72 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 ...
6
votes
2answers
1k 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
104 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
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0answers
107 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. ...
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1answer
146 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
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0answers
135 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
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1answer
194 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
664 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 ...
17
votes
2answers
5k 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
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1answer
263 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
207 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
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1answer
265 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
659 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 ...
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0answers
45 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
131 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 ...
5
votes
1answer
684 views

Showing the sum of orthogonal projections with orthogonal ranges is also an orthogonal projection

Show that if $P$ and $Q$ are two orthogonal projections with orthogonal ranges, then $P+Q$ is also an orthogonal projection. First I need to show $(P+Q)^\ast = P+Q$. I am thinking that since ...
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vote
0answers
291 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
0
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1answer
118 views

Tensor product of Hilbert Algebras

A Hilbert algebra is an inner product space that is also a *-algebra where the various operations and structures interact according to some axioms. One of those axioms is that the linear operation ...
0
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1answer
160 views

Averaging differential forms

Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
5
votes
3answers
849 views

Question about example of non-separable Hilbert space

I have come across the following example of a non-separable Hilbert space: Example 2.84. Let $I$ be a set, equipped with the discrete topology and the counting measure $\lambda_{\text{ count}}$ ...
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1answer
142 views

Hilbert spaces other than $L^2$

From measure theory we know that if $G$ is a finite measure space then $p \leq p^\prime$ implies $L^{p^\prime}(G) \subset L^p(G)$ where $L^p$ is the space of all $p$-integrable functions. So let $G$ ...
4
votes
5answers
945 views

Subspaces of Hilbert Spaces of finite dimension

Given a Hilbert space $H$ of finite dimension, why is any subspace of this space closed? I tried bashing out an answer using an arbitrary Cauchy sequence $\{ f_1 , f_2, \ldots \} \subset S \subset H $ ...
4
votes
2answers
154 views

Do all angles occur in Hilbert spaces?

Let $X$ be a Hilbert space with scalar product $(\cdot,\cdot)$. Then for two vectors $v,w$ of norm $1$, we can interpret $(v,w)$ as an angle, so that $(v,w)=\cos(\varphi)$ for a unique angle ...