“Some Results on the Dual Space of Normed Almost Linear Space”

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 .