In a generad normel linear space X R.C.James introduced the definition that x⊥y if and only if ‖x‖≤‖x+λy‖ for any λ∈Φ. Taking account of James' definition

we introduce the fellowing concepts. [Definition1.2] If X =MN and M⊥N

then N is Called the right orthogonal complement of M(in X)

denoted by M~⊥) M is called the left orthogonal complement of N(in X)

denoted by ~⊥N. In this paper we shall study the most fundamental questions of the orthogonal complements mentioned above

viz.; [1] the existence of orthogonal complements (§3); [2] the uniqueness of the or thegonal complements (§4) [3] the structural representation of the riht orthogonal complements (§5); [4] the relations between the right orthogonal complements and projections and the norm-preserved extension of linear operators (§2). We shall obtain some significant results; several of them generalize and improve the known theorems. Theny are [1] [Corollary 2.2] Suppose that X is a innerproduct space and p a projection of X (P≠0). Then P is the orthogonal projection if and only if ‖P‖=1. [2] [Exemple 3.6] There is a 3-dimensional Banach space

in which every 2-dimensional subspace M doce not possess the right orthogonal complement M~⊥;and consequently

every 1-dimensional subspace N does not possess the left orthogonal complement ~⊥N. [3] [Corollary 5.3] Suppose that X is a (complex) smooth normed linear space and M is a subspace of X. If {X_α|α∈Λ} is the totality of subspaces of X which contain M as a subspace of dimension 1

then the right orthogonal complement of M in X exists if and enly if M in every X_α exists. [4] [Theorem 6.1] In a continuous semi-inner product space X which is reflexive with respect to the norm induced by the semiiner product. Then

to every continuous linear functional f∈X* there exists a vector y∈X such that f(x)=[x

y] for all x∈X. Furthermore

if X is also strictly convex with respect to its norm

then y is unique.