Ruang l^p pada Norm-2 Lengkap

Abstract

The space l^p with 1≤p∞ is the set of real numbers that satisfy _(n=1)^∞▒〖|x_n |^p∞〗.The function in the vector space X which has real value which fulfills the norm-2 properties is denoted by ,⋅‖ and the pair (X,‖⋅,⋅‖) is called the norm-2 space.A norm-2 space is said to be complete or called a Banach-2 space if every Cauchy sequence in the space converges to an element in that space.This research was conducted to prove the l^p space in the complete norm-2.The first step to prove the completeness is to prove that the norm contained in l^p with 1≤p∞ satisfies the properties of norm-2.Next, prove that the norm derived from norm-2 is equivalent to the norm in l^p.Next shows that every Cauchy sequence in space l^p converges to an element in space l^p.Based on this proof, it is found that (l^p,‖⋅,⋅‖) is a complete norm-2 space.