# Unitary space

A vector space over the field $\mathbf C$ of complex numbers, on which there is given an inner product of vectors (where the product $(a,b)$ of two vectors $a$ and $b$ is, in general, a complex number) that satisfies the following axioms:

1) $(a,b)=\overline{(b,a)}$;

2) $(\alpha a,b)=\alpha(a,b)$;

3) $(a+b,c)=(a,c)+(b,c)$;

4) if $a\neq0$, then $(a,a)>0$, i.e. the scalar square of a non-zero vector is a positive real number.

A unitary space need not be finite-dimensional. In a unitary space one can, just as in Euclidean spaces, introduce the concept of orthogonality and of an orthonormal system of vectors, and in the finite-dimensional case one can prove the existence of an orthonormal basis.

#### Comments

#### References

[a1] | W. Noll, "Finite dimensional spaces" , M. Nijhoff (1987) pp. 338 |

[a2] | W.H. Greub, "Linear algebra" , Springer (1975) pp. Chapt. XI |

**How to Cite This Entry:**

Unitary space.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Unitary_space&oldid=31532