Uniform boundedness principle for bounded linear operators (Banach-Steinhaus theorem) is one of the cornerstones of classical functional analysis (see, e.g., [1-3] and the references cited therein).
The following uniform boundedness principle is the central result of this article.
Later chapters explain the Fourier transform, fixed point theorem, Baire category theorem, Hahn-Banach theorem, uniform boundedness principle
, spectral theory of operators on Hilbert spaces, weak topologies, and compactness in metric spaces.
According to eventually uniform boundedness principle
, we have the following conclusion: only [psi]([y.sub.1], [z.sub.1]) = (u([square root of [y.sup.2.sub.1] + [z.sup.2.sub.1]])/[square root of [y.sup.2.sub.1] + [z.sup.2.sub.1]]) - 1/4([[phi].sup.2.sub.1] + [[phi].sup.2.sub.2]) > 0 is satisfied; the system is uniform and stable boundedness ultimately.
A familiar technique in this respect is to apply Baire category theorem via the uniform boundedness principle
. This note displays arguments of a different kind, where the leading role is played by James' fundamental characterization of weak compactness [J], and Simons' inequality [S] which proves it in the separable case (see [Pf1], [Pf2] for recent and deep progress in the non-separable frame).