Let X be a smooth complex projective variety of dimension n and let L be an ample line bundle on X. In this paper, in order to investigate the dimension of H 0 ( K X + t L ) more systematically, we introduce the invariant A i ( X , L ) for every integer i with 0 ⩽ i ⩽ n . Furthermore, we study this invariant for the case where L is ample and spanned by global sections. As applications we get a lower bound (resp. an upper bound) for the dimension of H 0 ( K X + t L ) if L is ample and spanned by global sections (resp. very ample).