AN INNER PRODUCT THAT MAKES A SET OF VECTORS ORTHONORMAL HENDRA GUNAWAN Given a linearly independent set {a 1 ,...,a n } in a real inner product space (X, 〈·, ·〉) of dimension d ≥ n (d may be infinite), an average undergraduate student can do Gram-Schmidt process to obtain an orthonormal set {a ∗ 1 ,...,a ∗ n } from {a 1 ,...,a n }. Now, leaving the set {a 1 ,...,a n } as it is, how can we explicitly derive a new inner product 〈·, ·〉 ∗ from the given inner product 〈·, ·〉 such that, with respect to the new inner product, {a 1 ,...,a n } becomes an orthonormal set in X ? Basically, this can be done in the following way. Let S be the subspace spanned by {a 1 ,...,a n }, P be the projection on S and Q = I −P be its complementary projection. Next let T be a linear transformation that maps {a 1 ,...,a n } to an orthonormal basis for S and define a new inner product 〈·, ·〉 ∗ on X by 〈x, y 〉 ∗ := 〈TPx,TPy 〉 + 〈Qx, Qy 〉. (1) Then clearly we have 〈a i ,a j 〉 ∗ = δ ij for all i, j ∈{1,...,n}, that is, {a 1 ,...,a n } is orthonormal with respect to 〈·, ·〉 ∗ . To get an explicit formula for 〈·, ·〉 ∗ from (1), however, one needs to work out the expressions for P and T and then plug them to (1). Here P will be of the form P (x)= ∑ n i=1 α i a i with α i = α i (x, a 1 ,...,a n ),i =1,...,n. Meanwhile T can be represented by an n × n matrix, but one has to choose or, most likely, construct an orthonormal basis for S first (this can be done, e.g., by applying the Gram-Schmidt Keywords and phrases: inner products, orthogonality, n-inner products 2000 Mathematics Subject Classification: 46C50, 46C99, 15A03. 1