We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable maps, preserving adjoint morphisms and all fini...

this paper is concerned with the best proximity pair problem in hilbert spaces. given two subsets $a$ and $b$ of a hilbert space $h$ and the set-valued maps $f:a o 2^ b$ and $g:a_0 o 2^{a_0}$, where $a_0={xin a: |x-y|=d(a,b)~~~mbox{for some}~~~ yin b}$, best proximity pair theorems provide sufficient conditions that ensure the existence of an $x_0in a$ such that $$d(g(x_0),f(x_0))=d(a,b).$$

In this paper, we establish an existence result of the solution for an generalized strong vector variational inequality already considered in the literature and as applications we obtain a new coincidence point theorem in Hilbert spaces.    

In this paper, we study the existence of generalized solutions for the infinite dimensional nonlinear stochastic differential inclusions $dx(t) in F(t,x(t))dt +G(t,x(t))dW_t$ in which the multifunction $F$ is semimonotone and hemicontinuous and the operator-valued multifunction $G$ satisfies a Lipschitz condition. We define the It^{o} stochastic integral of operator set-valued stochastic pr...

When describing a quantum mechanical system, it is convenient to consider state vectors that do not belong to the Hilbert space. In the rst part of this paper, we survey the various formalisms have been introduced for giving a rigorous mathematical justiication to this procedure: rigged Hilbert spaces (RHS), scales or lattices of Hilbert spaces (LHS), nested Hilbert spaces, partial inner produc...

Variable Hilbert scales are an important tool for the recent analysis of inverse problems in Hilbert spaces, as these constitute a way to describe smoothness of objects other than functions on domains. Previous analysis of such classes of Hilbert spaces focused on interpolation properties, which allows us to vary between such spaces. In the context of discretization of inverse problems, first r...

Reproducing kernel Hilbert spaces and wavelets are both mathematical tools used in system identification and approximation. Reproducing kernel Hilbert spaces are function spaces possessing special characteristics that facilitate the search for solutions to norm minimization problems [3]. As such, they are of interest in a variety of areas including Machine Learning [11]. Wavelets are another mo...

We extend the standard Fourier multiplier result to square integrable functions with values in (possibly nonseparable) Hilbert spaces. As a corollary, we extend the standard Hardy class boundary trace result to H (even Nevanlinna or bounded type) functions whose values are bounded linear operators between Hilbert spaces. Both results have been well-known in the case that the Hilbert spaces are ...

Abstract. In this article, we study tensor product of Hilbert C∗-modules and Hilbert spaces. We show that if E is a Hilbert A-module and F is a Hilbert B-module, then tensor product of frames (orthonormal bases) for E and F produce frames (orthonormal bases) for Hilbert A⊗B-module E ⊗F , and we get more results. For Hilbert spaces H and K, we study tensor product of frames of subspaces for H an...

We address the reasons why the “Wick-rotated”, positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point o...