The classical Lagrange identity is the following:

This can be proven by expanding and separating the terms into the cross-terms part and the non cross-terms part.

The Lagrange identity implies the Cauchy-Schwarz inequality in . And when , this can be rephrased as

for . In general, the term can be identified as the norm squared of the wedge product .

In this note, we give the less well-known extension of this identity and the corresponding Cauchy-Schwarz type inequality.

Let be an inner product space and denote its inner product by “” or (whichever is clearer). The -fold tensor product space can then be naturally endowed with the inner product (extended linearly)

For simplicity, let us denote by .

Now consider the -fold wedge product space of . For , the inner product is defined by (extended linearly)

For example, for an orthonormal basis in . It turns out that if we identify with

where is the permutation group of objects, then . To see this,

Therefore (1) really defines an inner product. The extended Lagrange identity is then just the expansion

Here is the alternating subgroup of , which consists of all the even permutations.

The extended Cauchy-Schwarz inequality is just

When , (3) is just the classical Cauchy-Schwarz inequality

For , (2) gives

The Cauchy-Schwarz inequality then becomes

From (4) we also obtain an interesting inequality

with the equality case holds if and only if are orthogonal to each other or one of them is zero.

It seems that when , the algebra becomes quite formidable. Let us write . Then

The Cauchy-Schwarz inequality in this case is rather complicated:

Observe that some of the “odd permutation terms” (those with a negative sign in (6), e.g. ) are non-negative, so we can drop these terms and rearrange it to have an inequality similar to (5):

The equality holds if and only if these ‘s are orthogonal to each other or one of them is zero.