Definition of "euclidean"

If ( α 1 , … , α n ) {\displaystyle (\alpha _{1},\dots ,\alpha _{n})} is an orthonormal basis of a euclidean space V {\displaystyle V} , then for all α , β {\displaystyle \alpha ,\beta } in V {\displaystyle V} α = ∑ i = 1 n ( α | α i ) α i and ( α | β ) = ∑ i = 1 n ( α | α i ) ( β | α i ) . {\displaystyle \alpha =\sum _{i=1}^{n}(\alpha \vert \alpha _{i})\alpha _{i}\quad {\text{and}}\quad (\alpha \vert \beta )=\sum _{i=1}^{n}(\alpha \vert \alpha _{i})(\beta \vert \alpha _{i}).} We turn now to a somewhat more profound matter. Suppose that we partially order by inclusion the collection of all orthonormal sets of vectors in a euclidean space V {\displaystyle V} . A maximal element of this set is called a maximal orthonormal set.

1968, Robert R. Stoll, Edward T. Wong, Linear Algebra, London: Academic Press, page 76