Difference between revisions of "Unitary space"
From Encyclopedia of Mathematics
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| − | A [[Vector space|vector space]] over the field | + | {{TEX|done}} |
| + | A [[Vector space|vector space]] over the field $\mathbf C$ of complex numbers, on which there is given an [[Inner product|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) | + | 1) $(a,b)=\overline{(b,a)}$; |
| − | 2) | + | 2) $(\alpha a,b)=\alpha(a,b)$; |
| − | 3) | + | 3) $(a+b,c)=(a,c)+(b,c)$; |
| − | 4) if | + | 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. | 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. | ||
Revision as of 17:34, 11 April 2014
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://encyclopediaofmath.org/index.php?title=Unitary_space&oldid=17853
Unitary space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_space&oldid=17853
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article