where C is a nonempty closed convex subset of a real

Hilbert space H and f : H x H [right arrow] R is a given bifunction with f (x,x) = 0 for all x [member of] C.

Progress in brain research indicates (Roy and Kafatos, 1999, 2003, 2004) that to apply the wave function formalism used in quantum mechanics, it is necessary to consider relevant

Hilbert space structure.

Let H be a real

Hilbert space with inner product [??]*,* [??] and norm [parallel] * [parallel], respectively.

However, the discussions in [11, 12, 13] supposed that the kernel used to construct the reproducing kernel

Hilbert space is predefined.

A frame for a

Hilbert space firstly emerged in the work on nonharmonic Fourier series owing to Duffin and Schaeffer [1], which has made great contributions to various fields because of its nice properties; the reader can examine the papers [2-12] for background and details of frames.

where A is the infinitesimal generator of an analytic semigroup {S(t), t [greater than or equal to] 0} of bounded linear operators in a real separable

Hilbert space X.

In virtually any representation of quantum theory, the states can be perceived as constructed in a suitable user-friendly

Hilbert space H.

In the mathematical framework of information theory, density operators (quantum states) are positive operators with trace 1 on complex separable

Hilbert space H, and the set of all density operators is denoted by S(H) on H.

Throughout this paper, we always assume that C be a closed convex subset of a real

Hilbert space H with inner product and norm are denoted by <*,*> and [parallel]*[parallel], respectively.

[6] considered an implicit Volterra difference equation in a

Hilbert space and obtained sufficient conditions so that the solutions exist and have a bounded behavior.

(On the other hand, if a Banach space is a CAT([kappa]) space for some [kappa] [member of] R, then it is necessarily a

Hilbert space and CAT(0).) For a thorough discussion of these spaces and of the fundamental role they play in geometry, see Bridson and Haefliger [2].

A

Hilbert space over the complex field is denoted, in [6], as H.The standard definition of a norm on an inner product space, including on a

Hilbert space, is