# Tagged Questions

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### Checking whether an operator is self-adjoint. Problem with domain of an operator.

I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The ...
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I was trying to understand why this operator is hermitian (I see that) but not self-adjoint: We have the operator $P:\phi(\cdot )\mapsto -i\phi '(\cdot )$, dfined in $C_0^\infty (I)$ (infinitely ...
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### Condition for the existence a vector non-orthogonal to each vector of a given family?

Consider $k$ vectors $x_1,\dotsc,x_k$ of a $n$-dimensional Hilbert space. General conditions on $k$ and/or on the $x_j$'s for the existence of a vector $a$ such that $\langle a,x_j\rangle≠0$ for all ...
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### Questions about the proof of the Riesz representation theorem

Let $H$ be Hilbert space, $f:H \rightarrow \Bbb F$ linear and bounded map. I'm trying to prove that there exists only one $z_0 \in H$ such that: $\forall_{x \in H} : f(x)=\langle x,z_0\rangle$ ...
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### Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
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### Hilbert space continuous linear map, one dimensional subspace

Could you help me with the following exercise? Let $H$ be a Hilbert space, $\alpha : H \rightarrow \mathbb{C}$ a linear continuous mapping, $\alpha \neq 0$. Prove that the orthogonal complement ...
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### Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
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I got stucked in this problem and get no clue to solve this. Can any one please help me? Thanks Suppose $X$ is an inner product space. If for every bounded linear function $f$, there exists $z \in ... 1answer 163 views ### Orthonormal Basis for Hilbert Spaces The following is the definition of orthonormal base that I am using: The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space H, an ... 0answers 38 views ### How to find a hilbert basis of a given subspace considering a given inner product Let$X$be the space of continuous functions on$[-1;1]$to$\mathbb{R}$with the inner product: $$\langle f,\ g\rangle = \int_{-1}^{1} \! f(x)g(x) \, dx$$ and let$U$be a subspace of$X$with$U := ...
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I read the following proof that in a vector space $V$ of dimension $n$, a set of orthonormal vectors $\{\phi_1, ..., \phi_m\}$, with $m<n$, is not complete : Among the linear combinations ...
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Let $X$ be a Hilbert space with associated canonical isomorphism $I:X\rightarrow X^\ast$ (by the Riesz representation theorem). If $A:X\rightarrow X$ is a linear operator on $X$, then its dual ...
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### Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
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### What is a concrete example of a non-compact Hermitian operator on an infinite-dimensional Hilbert space whose eigenvectors do not form a complete set?

If I am not misunderstanding anything: by the spectral theorem, Hermitian operators that act upon finite-dimensional Hilbert space as well as compact Hermitian operators that act upon ...
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### If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
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### Showing that inner product of two vectors is the limit of the inner products

How can you show that $$(a,b) = (\sum_{i=1}^\infty a_i \phi_i,\sum_{i=1}^\infty b_i \phi_i) = \lim_{N\to \infty}(\sum_{i=1}^N a_i \phi_i,\sum_{i=1}^N b_i \phi_i)$$ where $\phi_i$ is an orthonormal ...
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### a question on decreasing sequence of subspaces

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
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### Basis means determinant of matrix of inner products is non-zero

Let $x_i$ be a basis of Hilbert space $X$ (NOT necessarily orthogonal) How do I show that $\text{det}((x_i,x_j)_H)_{ij} \neq 0$ for $i,j=1,...,n$? I see this fact used in Galerkin approximation ...
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### A strictly positive operator is invertible

Suppose that $H$ is an Hilbert space, and $T: H \to H$ is a self-adjoint strictly positive operator (i.e. $\langle Tx,x\rangle > 0$ for all $x \neq 0$). How do I show that this operator is ...
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### Prove that for every $f$ in $H$, the sequence $u_n$ which is the projection of $f$ on $K_n$ converges to a limit

$K_n$ is non-increasing sequence of closed convex sets in Hilbert space $H$ such that the intersection of $K_n$ is different from emptiness. Prove that for every $f \in H$, the sequence $u_n$ which is ...
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### 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 ...
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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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### 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 ...
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### 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 ...
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### Inequality of bounded linear operators on Hilbert space

Let $T$ and $S$ be bounded linear operators on a Hilbert space $H$. Verify that: $||TS||\leq ||T||\cdot ||S||$. The definition of the operator's norm is $||T||=\sup\{||Tv||_H: ||v||_H=1\}$. ...
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### Proof of normal operator and self-adjoint operator

1) Let $T∈L(V,V)$ be a normal operator. Prove that $||T(v)||=||T^*(v)||$ for every $v∈V$. ($T^*$ is the adjoint of $T$) 2) Let $T$ be an operator on the finite dimensional inner product space ...
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### Show that its a Generalized Eigenvalue problem

Show that the minimizer is obtained by a generalized eigenvalue problem. $$\alpha=\underset{1^TK\alpha=0; \ \alpha^TK^2\alpha=1}{\text {arg min}} \gamma ||f||_{K}^2+f^TLf$$ Details: $K$ ...
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### Reproducing Kernels are Positive Definite. Does the converse hold true?

Does the graph laplacian matrix $L$ form a reproducing kernel- given that the matrix is positive semi-definite. I was told in a hallway by a post doc- a month ago that the pseudo-inverse of $L$ forms ...
We know that any symmetric positive semi-definite matrix $K$ can be written as $K= AA^T$, where $A$ has real components. One way to get to $A$ is to compute eigen value decomposition of $K= P^T DP$ ...
Am trying to find a real scalar $\gamma$ such that for a given pair of real rectangular matrices $X,Y$ the following holds: $\frac{||Y||_{F}^{2}}{5} \leq ||\gamma X||_{F}^{2}\leq ||Y||_{F}^{2}$ ...