The inner product of the 4-acceleration and the 4-velocity is therefore always zero.

The inner product of two vectors is a complex number.

Equipped with this inner product, L is in fact complete.

The pairing between these two spaces also takes the form of an inner product.

The complex absolute value is a special case of the norm in an inner product space.

We can calculate the value of n by considering the inner product.

The relationship remains true independent of the frame in which the inner product is calculated.

This is slightly different than the above definition, which permits a change of inner product.

Two such vectors are adjacent when their inner product is 8.

This result is perhaps most transparent by considering the inner product defined above.

Let A be an operator on a finite-dimensional inner product space.

Some results only hold for inner product norms, however.

The set together with the energetic inner product is a pre-Hilbert space.

The inner product is linear in its first argument.

The bra operators are defined to be consistent with the inner product.

The inner product facilitates the construction of many useful concepts.

Any complete inner product space V has an orthonormal basis.

This in turn induces an inner product on the space of k-vectors.

Such an inner product will be bilinear: that is, linear in each argument.

Straightforward computation shows that the inner product is indeed preserved.

An important example is a Hilbert space, where the norm arises from an inner product.

The inner product of an element with itself is positive definite:

It is not specifically the inner product on a normed vector space.

For on , certain functions can be expressed as an inner product (in usually a different space).

Here, denotes the standard complex inner product on and .