“Some Results on the Dual Space of Normed Almost Linear Spaceâ€

Abstract

G. Godini introduced the concept normed almost linear space which generalizes normed linear space. To support the idea that the normed almost linear space is a good concept the notion of a dual space of normed almost linear space X, has been introduced in this paper. In this paper we prove some results like if X is normed almost linear space then i)   is a Banach space,  ii) If B is a basis of  X then for each B\  there exists f such that f( ) =1 and f( ) = 0  for each b  B\{ }. If then f iii) If has a basis, then X {0} iv)If  B is a basis such that card(B\ ) ∞, then X^* =  { f :  f \ ( )^*} and is total over X andv) If f ( )^*},then there exists  X^* such that \  = f, ||| ||| =    ||| f ||| and \ =0. Using these results we prove that  if X is strong normed almost linear space such that  is a metric and if x  X\ ( +  ), X ={ x₀ + (-x₀) +w+v : w ,  v ,  , 0} then  i) for each  f  ( )^* there exists  such that  \ = f  ii) V_X* {0} and ii) for eachf (W_X+V_X)^* there exists  such that   \ ) = f .