# Tagged Questions

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

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### Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1$ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
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### Prove or disprove existence of a sequence converging weakly to $0$ in an infinite dim Hilbert space

This is a problem on an old analysis qual, the prompt is: "Prove or give a counter example: if $H$ is an infinite dimensional Hilbert space and $0$ is the zero vector in $H$, then there exists a ...
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### Bounded operator and Compactness problem

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a ...
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### Space of Jordan curves

The space of square-integrable functions $f:[0,1]\rightarrow\mathbb{R}$ is well conceivable: it's essentially an $\infty$-dimensional Euclidean space (the Hilbert space $L^2$) with well interpretable ...
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### Example: operator injective, then the adjoint is NOT surjective

Let $T: V \rightarrow W$ be a bounded operator on normed spaces $V,W$. Now, there is a unique adjoint operator $T': W' \rightarrow V'$ defined by $T'(\alpha) = \alpha \circ T$. In finite dimensional ...
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### Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$dB_t, \; B_0 = 0,$$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times [0,1]$....
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### Composition of projections has a fixed point in a Hilbert space

Let Let H be a Hilbert space with an inner product ⟨⋅,⋅⟩ : H×H→R, and induced norm $∥⋅∥ : H→R_+$ Let $C_1$ and $C_2$ be closed, convex, nonempty, disjoint subsets of $H$ with at least one of ...
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### Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
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### Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$-\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0$$ on the bounded open set $U\subset\mathbb{R}^n$ ...
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### Reproducing Kernel Hilbert Spaces for Dummies

I am in the middle of some machine learning paper that states that for function $f$, imposing the norm constraint, $\|f \|=1$, corresponds to an orthogonal projection onto the direction selected in ...
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### 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 ...
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### Bounded Operators: Topological Dual

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider the bounded operators: $$\mathcal{B}(\mathcal{H},\mathcal{K}):=\{T:\mathcal{H}\to\mathcal{K}:\|T\|<\infty\}$$ Regard the linear ...
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### Comparing two sigma algebras in Hilbert spaces

Let $H$ be a non-separable Hilbert space. We denote $B$ by the sigma algebra generated by the norm topology in $H$. We also denote $B_{w}$ by the sigma algebra generated by the weak topology in $H$. ...
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### Every complete orthonormal set in a Hilbert space $H$ is an orthonormal basis, if and only if $H$ is finite dimensional.

Show that any orthonormal set in a Hilbert space $H$ is linearly independent, and use this to show that $H$ is finite dimensional if and only if every complete orthonormal set is an orthonormal basis. ...
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Definitions Given an operator algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ with $1\in\mathcal{A}$ Consider selfadjoint operators $A=A^*\in\mathcal{A}$. Define positive elements by: $$A\... 0answers 76 views ### In which cases the spectrum of an operator contains only eigenvalues? Let X\neq \{0\} be a complex normed spaces (not necessarily finite-dimensional) and T:D(T)\subset X\to X a linear operator (not necessarily bounded). I would like to know under what conditions can ... 1answer 89 views ### In a separable Hilbert space, can you write an operator from \mathcal H to \mathcal H as a column-finite matrix? In this question, we are representing an operator T as a matrix with respect to an orthonormal basis \left\{e_n : n \in \mathbb{N}\right\}. To do so, we let t_{ij} = \langle T(e_j),e_i\rangle. ... 0answers 249 views ### Is there an orthonormal basis for L_2[0,1] consisting of convex functions? Is there an orthonormal basis \{\phi_{\alpha}\} for the space L_2[0,1] of square-integrable functions from [0,1] to \mathbb{R} such that every \phi_{\alpha} is convex? Edit: A helpful ... 1answer 221 views ### Prove or disprove this argument Let L>0 and let \Omega be the set of all integrable functions from [0,L] to ]0,+\infty[. For all \varphi, \psi \in \Omega define \left \langle \varphi,\psi \right \rangle:=\int_{0}^{L}\... 0answers 174 views ### Don't understand this proof of equivalence of weak solutions to PDE I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, \mathcal{V} = L^2(0,T;V)). See the very last line in the image for that part:$$$$I give here ... 0answers 124 views ### Is this a spectral decomposition/embedding/isometry? Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where G=X^TX and X=U\lambda^{1/2} via the eigen decomposition of G. Now if I take the same ... 1answer 232 views ### Conditions for the sequence being weakly convergent Let H=\ell_2 be the Hilbert space of the square-summable sequences where$$ \langle x,y\rangle=\sum_{i=1}^{\infty}x_iy_i, \quad \|x\|=\sqrt{\langle x,x\rangle}. $$Let F: H\rightarrow H be an ... 0answers 521 views ### Sum of operator and adjoint is self-adjoint In abstract Hodge theory there is the following lemma: Let H be a Hilbert space and A \in \mathcal{C}(H) a densely defined, closed operator (so possibly unbounded) and A^* its adjoint operator. ... 0answers 215 views ### Relations between spectrum and quadratic forms in the unbounded case Let H be a complex Hilbert space. If B is a bounded self-adjoint operator on H then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ... 1answer 232 views ### What are the negative-dimentional n-sphere and n-cube? The generalized formula for the volume and surface area of n-sphere allows to evaluate volumes and areas of negative-dimentional n-spheres.$$\begin{array}{ll} S_{n-1}(R) &= \displaystyle{\frac{n\...
Let $H$ be a pre-Hilbert space such that any closed sub space $M \subset H$ has the property $M^{\bot \bot}=M$. Prove that $H$ is a Hilbert space (ie, prove that $H$ is complete) My attempt: As \$H^*...