In this note, we assume that F is a linear space over the real or complex number field. The concept of semi-inner product space was introduced in 1961 by G. Lumer [8] but the main properties of it were discovered by J.R. Giles [9], P. L. Papini [10], P. M. Milicic [11], I. Rosca [12], B. Nath [13] and others. In this paper, we give the definition of this concept and point out the main facts which are derived directly from the definition.