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Questions tagged [hilbert-spaces]

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

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$x_0+M\subset K$ for some linear subspace $M$ of a Hilbert space implies $x_0$ is orthogonal to $M$, where $K$ is a closed convex set

Suppose $K$ is a closed convex set in a Hilbert space $H$. If $x_0+M\subset K$ for some linear subspace $M$ of $H$, prove that $\langle x_0,y\rangle=0$ for all $y\in M$; in other words, $x_0$ is ...
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17 views

Isomorphism between tensor product of Hilbert spaces

While I am familiar with the isomorphism between finite-dimensional vector spaces with the same dimension and isometric isomorphism between Hilbert spaces, I realize I am getting confused by $n$-th ...
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1answer
20 views

Showing that an integral operator is bounded on a hilbert space.

Im trying to understand how it was hsown that T is a bounded operator on H. Going through the proof I don't understand what justifies the convergence of the last integral in this proof? What is it ...
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27 views

Aupetit B, A primer on spectral theory, Exercise IV.1

Aupetit B, A primer on spectral theory, Exercise IV.1 Prove that $M_n\mathbb{(C)}$ and $L\mathcal{(H)}$ where $\mathcal{H}$ is a Hilbert space, have no characters My attempt: I've done $M_n\mathbb{...
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1answer
58 views

Prove that subset of Hilbert space is compact

I encountered the following difficulty while reading a proof in Bosq's Linear Processes in Function Spaces. Let $(v_j)_{j\geq 1}$ be an orthonormal basis in the real Hilbert space $H$. Let $1=M_1<...
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1answer
22 views

Density problem involving embedded Hilbert spaces

It is well knon that if $s, t\in\mathbb{R}$, with $s >t$, the following continuous embedding holds $$ H^s(\mathbb{R}^n)\hookrightarrow H^t(\mathbb{R}^n).$$ My question is: these spaces are also ...
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1answer
15 views

If $V$ is right-orthogonal, does it hold $\langle AV,BV\rangle_F=\langle A,B\rangle_F$?

Let $A,B\in\mathbb R^{m\times n}$. It's easy to see that for the Frobenius inner product it holds $$\langle A,B\rangle_F=\operatorname{tr}B^\ast A=\operatorname{tr}A^\ast B.\tag1$$ So, if $U\in\mathbb ...
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1answer
21 views

bounded operators on a hilbert space represented by a matrix using an orthonormal basis

It is written that a bounded operator, A, on a separable infinite dimensional Hilbert Space, H, can be represented by an infinite matrix using an orthogonal basis, $Ī±_{ij}= (Ae_j,e_i)$, ...
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1answer
25 views

Prove there is a unitary vector such that $\|T\| = \langle x,Tx \rangle$ [closed]

Let $H$ be a Hilbert space and $T$ be a compact operator in $B(H)$. If $T$ is positive, how we can find a unitary vector $x$ in $H$ such that $\| T \| = \langle x,Tx \rangle$? how we can find a ...
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1answer
12 views

Range of a unitary transformed orthogonal projection

Let $X$ be a Hilbert space, $P$ an orthogonal projection in $X$ and $Q \in L(X)$ (i.e. $Q \colon X \to X$ is linear and continuous) a unitary linear transformation, i.e. $Q^*Q=Id_X= QQ^*$ ($Q^*$ ...
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49 views

Property of the inner product of Hilbert space

Let $H_1\subset H_2\subset H_3$ be three continuosly embedded and dense complex Hilbert spaces, where $H_3$ is the dual space of $H_1$ and $A:H_1\longrightarrow H_2$ and $B:H_1\longrightarrow H_3$ be ...
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1answer
19 views

Rank of a left-orthogonal decomposition $A^T=UB^T$

I've got a rather simple question, but couldn't find an answer to it: Say $A\in\mathbb R^{m\times n}$ can be decomposed according to $$A^T=UB^T\in\mathbb R^{n\times m}\tag1$$ for some left-orthogonal (...
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54 views

When do three closed balls have a nonempty intersection?

Consider a real Hilbert space $\mathcal{X}$. For $(c,\rho)\in\mathcal{X}\times \mathbb{R}$, I denote the closed ball $B(c;\rho) = \{x \in \mathcal{X}|\|x -c\|\leq\rho\}$. I am curious if y'all know of ...
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1answer
41 views

Does $\|A\|$ or $-\|A\|$ belongs to $\sigma(A)$ for a normal operator

For a normal operator $A$ on a Hilbert space $H$, I know that spectral radius $r(A)=\|A\|$. The question says is it true that either $\|A\|$ or $-\|A\|$ belongs to spectrum $\sigma(A)$? I have ...
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1answer
19 views

Is there a proper ideal of $B(H)$ that contains a proper projection

Let $H$ be a infinite-dimensional separable Hilbert space and $\mathcal{I}$ be a proper closed two-sided ideal of $B(H)$. Can $\mathcal{I}$ contain a projection for a infinite dimensioal proper closed ...
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Show that $X$ is complete.If Y consist of all x in $X$ such that x(a)=0. Is Y a subspace of X? Do all x in$X$ of degree 2 form a subspace of X? [closed]

This question is already asked here. But i did not understand to solve this question. Please give a detailed answer with calculations.
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3answers
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Does $A^TA=I$ imply $AA^T=I$?

Let $m,n\in\mathbb R^{m\times n}$ with $$A^TA=I_n\tag1.$$ I wonder whether this implies that $$AA^T=I_m\tag2$$ or if we can show it at least in the case $m=n$. EDIT: I was hoping for a proof which ...
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2answers
22 views

Determining operator by inner product

It's been a while since I did any functional analysis. I am looking for a result that states the following. Suppose we are given an inner product space $H$ and two linear operators, $A$ and $B$. ...
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1answer
10 views

Inner Product-Norm Inequality

I am doing some work and reached an inequality where I think I should be able to go further but I am not sure how. I have a Hilbert space of functions on a set $X$. What I have: $$\forall x, y \in X, ...
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1answer
29 views

The classical projection theorem

I am going over a proof of the classical projection theorem which states the following: Let $H$ be a Hilbert space and $M$ a closed subspace of $H$. Corresponding to any vector $x \in H$, there is a ...
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1answer
47 views
+50

Non-monotone sequence of orthogonal projections

Let $\mathcal{H}$ be a separable Hilbert space and $A_m:\mathcal{H}\to \mathbb{R}^m$ linear with $\sup_m \|A_m\|<\infty$. Assume that for all $x\in\mathcal{H}\setminus \{0\}$ there is a $\...
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1answer
33 views

Please define $\ell^2$ space and give some examples of its subspaces

I have googled alot about $\ell^2$ space and its subspace. There are showing result for $L^2$. Is (small $\ell$) $\ell^2$ space and (greater $L$) $L^2$ space are same?
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Is the canonical isomorphism $\mathbb R^m\otimes\mathbb R^n\to\mathbb R^{m\times n}$ isometric with respect to the Frobenius norm?

Let $p\in\mathbb N$ and $n_1,\ldots,n_p\in\mathbb N$. Assume the tensor product space $\bigotimes_{i=1}^p\mathbb R^{n_i}$ is equipped with the unique inner product $\langle\;\cdot\;,\;\cdot\;\rangle_{\...
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1answer
19 views

Best approximation of a function among closed linear manifolds

Let $H$ be an infinite-dimensional Hilbert space and consider a $n-dimensional$ closed linear manifold generated by a subset of orthonormal basis, say, $M = span(\{u_1,u_2,\cdots,u_n\})$. Of course, ...
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+100

Exercise 5.22 in Brezis, “Functional Analysis Sobolev Spaces and Partial Differential Equations”.

Let $H$ be a Hilbert space, $C\subseteq H$ a nonempty closed convex set and $T:C\to C$ a nonlinear contraction, that is $$ (*)\qquad|Tu - Tv| \leq |u-v|. $$ Let $(u_n)$ be a sequence in $C$ such that $...
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13 views

Maxima of isonormal Gaussian process

Let $\mathcal{H}$ denote a real separable Hilbert space and let $W\colon \mathcal{H} \to \mathbf{R}$ denote an isonormal Gaussian process; that is $\mathbf{E}[W(t)W(s)] = \langle t, s \rangle_{\...
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24 views

I can say that : according to the previous lemma we have: $ \| u_k \|_2\leq 2 \| F_k (f_n ^ k) \|_2, \qquad \forall n \geq 1.$

Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $ X $ be a Banach space and $ H $ a Hilbert space. For $ t \in E $, we set $ F_a (f)(t) = f (t) 1_{\| f \| \leq a} (t)$ Lemma: Let $ \{...
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1answer
39 views

If $(U,\Sigma,V)$ is a singular value decomposition of $A$, do the first $\text{rank}A$ columns of $V$ and $U$ form orthonormal bases?

Let $m,n\in\mathbb N$ $A\in\mathbb R^{m\times n}$ and $|A|:=\sqrt{A^TA}$ $r:=\operatorname{rank}A$ $\sigma_1>\cdots>\sigma_r>\sigma_{r+1}=\cdots=\sigma_n=0$ denote the singular values of $A$ ...
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10 views

Norms and triangular inequality in Hilbert spaces

My question is about an intuition and it arises from the following problem. Let $C$ be a convex set (nonempty and closed) in $R^d$, and $P_C$ be the orthogonal projection onto $C$. We need to show ...
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3answers
22 views

If $Z_n = X_n + Y_n$ for $X_n\in M$ and $Y_n\in N$ then $(X_n)$ and $(Y_n)$ converge

Let $H$ be a Hilbert space (infinite dim) with $M,N\subset H$ being closed subspaces satisfying $N\subset M^\perp$. I'm trying to show that $M+N$ is closed. If $(Z_n)_{n=1}^\infty \subset M+N$ is a ...
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15 views

Truncated singular value decomposition and error matrix

Let $m,n\in\mathbb N$ $A\in\mathbb R^{m\times n}$ $r:=\operatorname{rank}A$ $\sigma_1>\cdots>\sigma_r>\sigma_{r+1}=\cdots=\sigma_n=0$ denote the singular values of $A$ We say that $(U,\...
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1answer
24 views

Compute the orthogonal complement of a vector space [closed]

I have the following question. In the Hilbert space $l^2$, compute the orthogonal complement of the vector space $X=\lbrace x=(x_n)_n\in l^2:x_{2n}=0 \ and \ x_1=x_3\rbrace $.
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1answer
26 views

Compute the adjoint of an operator in Hilbert space

I have the following question. In the Hilbert space $l^2$, consider the operator $Tx=(\frac{x_n+x_{n+1}}{2})_n$ and $x=(x_n)_n$. Compute the adjoint of operator $T$. I tried to find $T^*$ such that $(...
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1answer
27 views

Sum of orthogonal projection operators

Let $X$ be an Hilbert space, I am trying to see that if $P$ and $Q$ are orthogonal projection operators then the following are equivalent: $(1) Im Q\subseteq KerP$ $(2)P+Q$ is an orthogonal ...
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4answers
74 views

Limit $\lim_{\epsilon \rightarrow0} \frac{1}{\sqrt{\epsilon}}\exp\left({-\frac{x^2}{\epsilon}}\right)$ in the sense of distributions [closed]

Compute the following limit: $$\lim_{\epsilon \rightarrow0} \frac{1}{\sqrt{\epsilon}}\exp\left({-\frac{x^2}{\epsilon}}\right)$$ I do not know how the "sense of distributions" is applied in this ...
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1answer
13 views

Proving a linear form to be continuous

In the Hilbert space $L^2(R)$ I have seen that the following form is linear, however, I need to check if it is continuous and find the associated vector using the Riesz-FrƩche Theorem. I have tried to ...
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28 views

Upper and lower bounds for the Grothendieck inequality

The Grothendieck inequality states that there is a universal constant ${\displaystyle K_{G}}$ with the following property. If $M_{i,j}$ is an $n$ by $n$ (real or complex) matrix with $$\left|\sum_{i,j}...
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1answer
25 views

Representation of diagonal operator in $l^2$ as a multiplication operator in $L_2(X,\mu)$ (spectral theorem)

There is a (weaker) version of spectral theorem saying that any self-adjoint operator in Hilbert space is unitarily isomorphic to multiplication operator in $L_2(X,\mu)$, where $(X,\mu)$ is some ...
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19 views

Show that if $A$ is a symmetric operator then its domain is a subset of domain of its adjoint operator

Let $A \in L(D,H)$ and $A$ is symetric operator show that $D \subset D(A^*)$. My attempt: Domain of $A^*$ is defined as follows Let $v\in H$, consider a linear form $\phi_v \in L(D, \mathbb{C}), \ ...
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1answer
28 views

Let $U \in U(H)$ be unitary operator on Hilbert space $H$. Is it possible for it to has an empty point spectrum?

Let $U\in U(H)$ be unitary operator. Is it possible for it to has an empty point spectrum? I am aware that every bounded operator acting on complex Hilbert space has non-empty spectrum. Since $\...
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0answers
28 views

Prove that if $\sum\|e_n-f_n\|^2<\infty$, then $\{f_n\}$ is complete. [duplicate]

Let $\{e_n\}$ and $\{f_n\}$ be orthonormal sequences in a Hilbert space $H$ with $\|e_n\|=\|f_n\|=1$ and $$\sum_{n=1}^{\infty}\|e_n-f_n\|^2<\infty.$$ Prove or disprove: If $\{e_n\}$ is ...
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16 views

Isonormal Gaussian process associated with a Hilbert space.

We consider the isonormal Gaussian process $W=\{W(h),h\in H\}$ indexed by a separable Hilbert space $H$, defined on a complete probability space $(\Omega, \mathcal F,P)$ where $\mathcal F:=\sigma(W)$,...
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1answer
13 views

Eigenvectors of hermitian matrix form unitary matrix

Let $A \in M_2(\mathbb{C})$ be a hermitian matrix i.e. $A = A^*$. Suppose that $\lambda, \mu$ are eigenvalues correspoing to normed eigenvectors $\begin{pmatrix} v_1 \\ v_2 \end{pmatrix}, \begin{...
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1answer
52 views

Spectrum perturbation

Let $A$ be a self-adjoint (not necessarily bounded) operator on a Hilbert space $\mathscr{H}$ and $B$ be self-adjoint bounded. Then how would one show that $\sigma (A+B) \subseteq \sigma(A) +\sigma(B)$...
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24 views

Complexified of the adjoint is the adjoint of the complexified

Given a real Hilbert space H, I can construct the complexified space $H_{\mathbb C}$, which is now a complex Hilbert space. Similarly, given a (possibly unbounded) densely defined operator $A:D(A)\...
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2answers
56 views

Exercise of compact self-adjoint operator

'Let $H$ be a Hilbert space. Find all compact self-adjoint operators $T:H \rightarrow H$ such that $T^{k}=0$ with $k>0, k \in N$.' $ \ $ I have this idea. Consider $\lambda_n$ eigenvalue of T and ...
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1answer
189 views

Prove that linear operator A has a nonzero kernel.

H is a separable Hilbert space, E is an inseparable Hilbert space, A is a continuous linear operator from E to the space L (H) of continuous operators on H with an operator norm. $A:E\to L(H).$ Prove ...
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2answers
56 views

Exercise on orthogonality in a Hilbert space

Problem Let $V$ and $W$ two subspaces of a Hilbert space $H$, with $dim(V)= m -1$ and $dim(W) = m$, with $m \geq 1$. Prove that if $V=span\left\{e_1, ..., e_{m-1}\right\}$, then there exists a vector ...
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1answer
29 views

Is a representation $(H,\phi)$ of a simple C*-algebra $A$ always faithful?

Suppose that $A$ is a simple C*-algebra (i.e. there is no closed ideal $I\subset A$ such that $0\neq I\neq A$) and let $(H,\phi)$ be a representation. Can we conclude that $(H,\phi)$ is faithful, i.e. ...
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0answers
14 views

Formula for inner product in a RKHS

Given a kernel function $k$ for which a Reproducing Kernel Hilbert Space (RKHS) $H$ exists, can I write a formula how to compute the inner product of two functions in $H$? I am, of course, aware that ...

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