Question

Prove that a bilinear form is an inner product.

Answer 1

To prove that a bilinear form is an inner product, we need to show that it is linear in both variables, symmetric, and positive definite.

Step-by-step explanation:

A bilinear form is an inner product if it satisfies certain properties. To prove that a bilinear form is an inner product, we need to show that it is linear in both variables, symmetric, and positive definite.

First, we show linearity in both variables. Let's consider a bilinear form B defined on a vector space V. For any vectors u, v, and w in V, and any scalar c, B satisfies the following properties:

• B(u+v, w) = B(u, w) + B(v, w)
• B(cu, v) = cB(u, v)

To prove symmetry, we show that B(u, v) = B(v, u) for all u and v. Finally, to show positive definiteness, we need to prove that B(u, u) > 0 for all u != 0.