An Analytical Innovation of Hilbert Algebra via δ-Approach Structure

Abstract

This paper aims to study an approach space, we introduced a new definition of  Hilbert algebra which is called approach Hilbert algebra, in this work we need to  define  approach inner product space. We proved every metric space is  approach space. Also, we proved the direct sum and the quolient space are approach Hilbert algebra, proved there exists symmetric isomorphism between Approach Hilbert algeba and set of all bounded B( A), through this result we can approach Hilbert algebra is semi-simple and found orthogonal components of closed right and left ideals, we proved some properties in qutient and direct sum of approach Hilbert algebra