In this paper,the orthonormality in 2-normed spaces is discussed and Bessel s inequality in 2-normed spaces is deduced.

On the foundation of the conception of orthonormal basis in finite dimensional Euclidean space,this paper provides the theory of completely orthonormal system in infinite dimensional Euclidean space.

Consider the state linear system ∑(A,B,C),If the orthonormal basis _n,n∈N exist in the state space Z,it is proved that the solution of the differential Riccati equation is of general form,i.

In this note, we prove that if {Φ(x-k)k|k∈Z} is tight frame with bound 1 in V, then {Φ(x-k)|k∈Z} must be an orthonormal basis of V.

Another method to seek normal orthogonal basis;

By applying elementary transformation,this paper obtains a new method to search for a normal orthogonal basis in Euclid space, it also gives a procedure by which the fewer third elementary transformations are enough to realize this method.

By means of congruent transformation in matrix,the method of transforming real quadratic form into standard form and the method of normal orthogonal basis are given in this paper.

The fact that each finite subspace has a unique orthogonal in an Eucliden space is proved in virtue of standard orthogonal basis.

The characteristic vector of the real symmetric matrix should be found out, which is orthogonalized and normalized to a standard orthogonal base and is used as row vector to construct the transformation matrix P, so the P~(-1)AP can be made into diagonal matrix.