Osama Mustafa Babiker2017-08-062017-08-062015http://hdl.handle.net/123456789/4659Abstract In this thesis we study the concept of completespaces , We process this concept on some spaces (Normed spaces –Banach spaces – Hilbert spaces),discuss some fundamental theorem's by using the concept of completeness to support the concept of extension for linear functions (Hahn – Banach theorem's ) together with linear operators (uniform boundedness principle ) .we study open and closed linearsoperators as an Introduction to The closed graph theorem .We apply thesetheorems to solve and prove some problems and theorems, andfinally we consider calculus on Normed spaces.الرياضياتMathematical Statistics