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A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?

In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
UserA's user avatar
  • 597
4 votes
0 answers
174 views

Constant in trace theorem for balls

Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$ The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
user avatar
2 votes
0 answers
88 views

Dependence and $L^2$ projections of functions

tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function? Let $w$ be a density on $\...
shawn532's user avatar
2 votes
0 answers
170 views

finite dimensionality of a subspace of a Banach space

Let $H$ be the space of measurable functions on $(0,1)$ such that $$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$ Let $C>0$ be a constant. Suppose that $W \...
Ali's user avatar
  • 4,143
2 votes
0 answers
120 views

Closure of Laplacian

Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator $$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$ There are two ...
B.Hueber's user avatar
  • 1,171
2 votes
0 answers
325 views

Examples of RKHS that are "classical"

Among the so-called "classical" Hilbert spaces ($L^2$, Sobolev, Hardy, Bergman, etc.), which are very well-studied, which are RKHSs? It is easy to construct example of RKHSs by applying the ...
lost_analyst's user avatar
2 votes
0 answers
56 views

Existence of a suitable smooth kernel

Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
Ali's user avatar
  • 4,143
2 votes
0 answers
553 views

$\ell_\infty$-norm covering number of RKHS ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$

For any $\epsilon \in (0,1)$, let $N_\infty(\epsilon, \mathcal{H}, R)$ denote the $\epsilon$-covering number of the RKHS norm ball $\{f\in\mathcal{H}: \|f\|_\mathcal{H} \leq R\}$ with respect to the $\...
melatonin15's user avatar
2 votes
0 answers
73 views

A question on groupoids and measurable fields of Hilbert spaces

Suppose that we have the following data: $ \mathcal{G} $ is a locally compact Hausdorff groupoid, with its source and range maps denoted by $ s $ and $ r $ respectively. $ (\lambda^{x})_{x \in \...
Transcendental's user avatar
2 votes
0 answers
341 views

Trace class operators convergent series

On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as $$ K = \sum_{n,m =0}^{\infty} K_{n,m}...
Kinzlin's user avatar
  • 305
2 votes
0 answers
2k views

Orthogonal complements of intersections of closed subspaces

Let $H$ be a Hilbert space and $H_1, \cdots, H_n$ be closed subspaces of $H$. $\mathbf{Question}:$ Is it always true that the orthogonal complement $(H_1\cap\cdots\cap H_n)^\bot$ of the intersection ...
Entaou's user avatar
  • 285
1 vote
0 answers
922 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
Rajesh D's user avatar
  • 698
1 vote
0 answers
94 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 square-...
mikew's user avatar
  • 108
0 votes
0 answers
106 views

Extension of a Hilbert basis

The picture below is taken from this paper: http://real.mtak.hu/22877/. The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...
Gustave's user avatar
  • 617