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

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478 views

distributivity of tensor product and direct sum for Hilbert spaces

Before I ask my actual question about direct sums and tensor products of Hilbert spaces, let's first talk about direct sums and tensor products of vector spaces. We might define direct sums of ...
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33 views

Equivalent Definitions of the $L_2$ inner product.

If $g \in L_2(\mathbb{R})$, then we can define the $L_2$ norm to have the following relationship: $\|g\|_2^2 = \int_\mathbb{R} g^2$. If $A\subseteq \mathbb{R}$, then we can define the norm of $L_2(A)$ ...
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50 views

Canonical isometric isomorphism of $l_{\alpha}^{2}$

Let $\alpha \in \mathbb{R}$ and $l_{\alpha}^{2}$ the vector space of bi-infinite sequences $(x_{n})_{n\in \mathbb{Z}}$ such that $||x||_{\alpha}:=\sum_{n\in\mathbb{Z}} ...
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417 views

Orthonormal Basis of $L^{2}(0,1)$?

Are the functions $e_n := e^{i\cdot(2n+1)\cdot\pi\cdot x}$, $n \in \mathbb{Z}$ an orthonormal basis of $L^{2}(0,1)$? I suppose it is true, but I haven't been able to prove it myself yet.
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276 views

Weak convergence in $C[0,1]$

For a uniformly bounded sequence $(f_n)$ in $C[0,1]$, show that $f_n$ converges weakly to $0$ $\iff $ $\lim \limits_{n \to \infty} f_n(y) =0$ for all $y \in [0,1]$ Is the equivalence true if we do ...
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44 views

Continuity of bounded and convex function on Hilbert space

I'm looking for the proof (or at least hints how to prove it) of theorem: Let $H$ be a real Hilbert space and let $f:H\to (-\infty,+\infty]$ be a convex function, bounded from above in some ...
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62 views

Equivalent Definitions of Negative Order Sobolev Spaces

Ignoring fractional sobolev spaces, if we restrict ourselves to $k>0$ when $k$ is an integer, then the Sobolev space of order $k$, for $W^{k,p}(\mathbb{R})$ is the space of functions $f$ such that ...
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367 views

Optimization problem using Reproducing Kernel Hilbert Spaces

I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernce to this paper [Olivier Chapelle, ...
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75 views

Fractional Brownian motion---construction via Hilbert space?

The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms: Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...
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1answer
77 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
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62 views

Any finite-dimensional subspace of a Hilbert space is closed: easier proof?

A noted theorem is that a finite-dimensional subspace of a Hilbert space must be topologically closed. I have seen some proofs of this theorem which are less simple than this, but what is wrong with ...
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26 views

Question about convergence of sum

Let $T\in B(H,E)$ where $H$ a seperable hilbertspace, $E$ a seperable Banach space. By parsevals identity $$\left\|T^*\right\|^2= \sup_{ \left\|x^*\right\|\leq 1}\left\|T^*x^*\right\|^2 = \sup_{ ...
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45 views

Bound on number of mutually orthonormal eigenfunctions

Let $E$ be the vector space of real valued continuous functions on an interval $[a,b]$. Let $K = K(x,y)$ be a continuous function of two variables, defined on the square $a \leq x \leq b$ and $a \leq ...
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76 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
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55 views

Is this subspace dense in $L^{2}(\Omega,\mu)$

Let $(\Omega,\mu)$ be a measure space, and let $X=L^{2}(\Omega,\mu)$ be the complex Hilbert space of square-integrable complex measurable functions on $\Omega$. (Each $f \in L^{2}$ is an equivalence ...
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84 views

Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$

Let $X$ be a complex Hilbert space, and let $A$ be a bounded linear operator on $X$. Define the real part of $A$ to be $\Re(A)=\frac{1}{2}(A^{\star}+A)$, and define ...
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71 views

$L$-Lipschitz gradient of $f$ implies inverse strongly monotone of $\nabla f$

Let $f:\; \mathcal{H} \to R$ be a continuously differentiable convex function such that $$\|\nabla f(x) -\nabla f(y)\|\leq L\|x-y\|.$$ Prove that the mapping $\nabla f$ is $1/L$ inverse strongly ...
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39 views

Subspace of certain series in a Hilbert space is compact

Let $E$ be a Hilbert space and let $\{x_{n}\}$ be an orthonormal basis.  Let $\{c_{n}\}$ be a sequence of positive numbers such that $\sum c_{n}^{2}$ converges.  Let $C$ be the subset of $E$ ...
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149 views

Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
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62 views

Find the norm of functional

Consider the functional from $l_2$. $$ x=(x_n)\mapsto \sum \frac{x_n+x_{n+1}}{2^n}. $$ What is the norm of the functional?
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72 views

Convergence of a sequence of unit vectors in a Hilbert space

Let $E$ be a vector space (over $\mathbb{R}$) with a positive definite hermitian form and let $\{x_{n}\}$ ($x_{n} \not= 0$) be a sequence converging to $x$ in the $L^{2}$ norm ...
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169 views

Basic Quantum Mechanics Concepts with Continuous Spectra

The following are a couple excerpts of the first chapter of Sakurai and Napolitano, Modern Quantum Mechanics, 2nd edition: Prior to these formulas, the text discusses the fundamental mathematics ...
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58 views

Sufficient condition for two operators being identical on Hilbert space

Considering two bounded linear operators $S,T$ in $\mathcal{B}(X)$, where $X$ is a complex Hilbert space. If $\def\norm#1#2{\langle {#1},{#2}\rangle} \norm{Sx}{x} = \norm{Tx}{x}$ for all $x\in X$, do ...
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128 views

Trivial Orthogonal Complement $\implies$ Denseness!

Disclaimer: Though I don't need it anymore this is interesting in its own! Is it true that if the orthogonal complement is trivial then the subset was dense: $$A^\bot=\{0\}\implies\overline{A}=X$$ ...
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132 views

What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
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59 views

Is projection on a convex closed weakly-sequentially continuos?

I think to have proved the following: Given K a convex closed(maybe also limited is needed)subset(also curve not just subspaces) of an Hilbert space H, is well defined the projection operator $p_K:H ...
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82 views

Please check proof of convergence in $L^2(0,T;L^2)$ of a composition (uses Nemytskii operator)

Suppose we have a continuous function $g:\mathbb{R} \to \mathbb{R}$ that satisfies $|g(x)| \leq C|x|$. Let $u_{n} \to u$ in $L^2(0,T;L^2)$. I want to show that $g(u_{n'}) \to g(u)$ in $L^2(0,T;L^2)$. ...
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107 views

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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30 views

subsets of closed ball of Hilbert space [closed]

Let the sequences $(x_n)$ and $(y_n)$ be subsets of unit closed ball $\overline{B}(0,1)$ of Hilbert space H. If $\lim_{n\to \infty}\langle x_n,y_n \rangle=1$, then $$\lim_{n\to ...
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20 views

I need help showing something is a linear continuous operator.

Define $T:C([0,1])\rightarrow C([0,1])$ by $T(f)(x)=f(0)+\int_0^xtf(t)dt$ I want to show that $T$ is a continuous linear operator.Showing the linear part is easy enough, but I am not quite sure how to ...
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71 views

Concluding that a linear operator on a Hilbert space is invertible

Setting: Let $H$ be a Hilbert space with two inner products, $\langle \cdot,\cdot\rangle$ and $[\cdot, \cdot]$, and $S:H\to H$ be a bounded linear operator such that for all $x,y\in H$, we have ...
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32 views

About what happens to eigenspace under functional calculus for Unbounded Operator

Let $T$ be an unbounded self adjoint positive operator on a Hilbert Space $\mathcal{H}$. Let $x \in \mathcal{H}$ be a vector such that $Tx = x$. Is it true that $T^{\frac{1}{2}} x = x$. For what $f$ ...
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96 views

Extending a compact operator to the entire Hilbert space

In a course I'm taking we defined compact operators as a linear mapping $H\rightarrow H$, where $H$ is a Hilbert space, that maps bounded sets to relative compact ones. The lecturer mentioned that the ...
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53 views

Proof: adjoint map of projection is a projection and …

Let $V$ be a pre hilbert space and $\pi \in \mathrm{End}(V)$. Show: the adjoint map $\pi^+$ of a projection (meaning: $\pi^2 = \pi$) is a projection itself. Show then: a projection $\pi$ is ...
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53 views

Isomorphism between Hilbert spaces

I want to show that the function $$ L^2(\Omega,\mathcal{O})\longrightarrow L^2(\widetilde{\Omega},\mathcal{O}) \colon f \longmapsto f|_{\widetilde{\Omega}}$$ is a isomorphism, where ...
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50 views

Determining the exact form of a projection in a Hilbert space

Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$ where $\mathcal{L}^2[0,T]$ is the set of Lebesgue ...
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122 views

Find the minimum distance that equal maximum inner product

If $x_0 \in$ $H$ (Hilbert Space) and $M$ is a closed linear subspace of $H$, prove that $$\min \{\|x - x_0\|: x \in M\} = \max \{\langle x_0, y\rangle : y \in M^\perp, \|y\| = 1\}.$$ I suppose $P$ ...
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105 views

Prove that this space is not Banach

Let $\Omega\subset\mathbb{R}^n$ be an open, bounded set with boundary $\partial\Omega$ of class $C^1$. $$\mathcal{A}:=\{u\in C^2(\bar\Omega):u=0\text{ on }\partial\Omega \}$$ endowed with the scalar ...
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89 views

Normal Compact Operator: not diagonalizable!

To proposition 5.17 in Weidmann's 'Lineare Operatoren in Hilberträumen' (german version) it is noted that the expansion of compact operators that are normal rather than self adjoint doesn't apply in ...
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45 views

Verification of conclusions regarding duality maps

I have two conclusions drawn from two results. I want to know how valid these two conclusions are. Firstly Consider the duality mapping(set-valued) $J:X \rightrightarrows X^{*}$ defined: $J(u) := \{ ...
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174 views

Symmetric Operator vs. Real Spectrum

For symmetric operators one has a characterization: $$A\text{ symmetric}:\quad A=A^*\iff\sigma(A)\subseteq\mathbb{R}$$ (I want to investigate to what extend symmetry is a necessary assumption.) ...
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57 views

When $M$ is closed $M^\perp$ is one-dimensional vector space

If $M=\{ x: Lx=0\}$, where $L$ is continuous linear functional on $H$ (Hilbert Space). Prove that $M^\perp$ is vector space of one-dimensional unless $M= H$. I know $M$ is closed so that $M^\perp$ is ...
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119 views

Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
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55 views

Maximal set at Hilbert space

I need to prove the following: If B orthonormal set in a Hilbert space X, then B is maximal if and only if $ B^{\perp} =\{0\} $ I tried the following : $ B^{\perp} =\{0\} $ => B maximal $ ...
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41 views

$T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$

The question goes as follows: $T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$. Given is the data: $X$ is a Hilbert space with an orthonormal ...
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73 views

A question on functional analysis

Let $H_i$, where $i = 1,2$ be Hilbert spaces and $T_i : H_i \rightarrow H_i$ be closed operators, such that $T_i$ have positive spectrum. Let $\phi : H_1 \rightarrow H_2$ is an isometric isomorphism ...
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40 views

$Te_n$ converging to zero

I have the following question in my functional analysis book I dont understand: $X$ is an infinite dimensional Hilbert space with an orthonormal basis $(e_n)$. Show that if $T \in K(X)$, then $Te_n ...
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39 views

Spectral theory - how to prove this lemma?

in Anver Friedman, Foundations of Modern Analysis I found a lemma (6.7.3): If A is a self-adjoint operator and $\{E_\lambda\}$ is a spectral family such that $A=\int_m^{M+\varepsilon} \lambda ...
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686 views

Square root of a Hermitian operator exists

There are a lot of questions here about square root operators, but none of them addresses the basic question of existence, and I didn't find a very beefy section in Wikipedia talking about this, so ...
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43 views

Proving $||f+g||\cdot||f-g|| \le ||f||^2+||g||^2$ in a Hilbert Space

Let $f$ and $g$ be vectors in a Hilbert space $H$. Show that $$||f+g||\cdot||f-g|| \le ||f||^2+||g||^2$$ My question is, do i have to rewrite $||f+g||$ as $\sqrt{\langle f+g,f+g\rangle}$ and same ...