A theory of summability on a space of generalized functions

A theory of summability of orthonormal sets is introduced in multinormed spaces. the approach which is presented caters for infinite sets { φ j } j ∈ j,where the index set j may be uncountable,and is applied to obtain convergence results in appropriate spaces of test functions and corresponding spaces of generalized functions. these spaces are constructed in a systematic manner that relies heavily on properties of orthonormal bases in hilbert spaces. a space of almost-periodic generalized functions,in which each generalized function can be expanded in terms of an uncountable basis of exponential functions,is obtained as a special case of our theory. © 2013 khaula naeem khan and wilson lamb.