Definition of a bilinear form:
A bilinear form is a function that takes two vectors as input and returns a scalar. In the context of a vector space V over a field F, a bilinear form is a function f: V × V → F that satisfies the following properties:

1. Linearity in the first argument: For all vectors x, y, z in V and all scalars a in F, f(ax + y, z) = af(x, z) + f(y, z).
2. Linearity in the second argument: For all vectors x, y, z in V and all scalars a in F, f(x, ay + z) = af(x, y) + f(x, z).

Proof that f is a bilinear form:

Linearity in the first argument:
Let's consider vectors x, y, z in V and a scalar c in F. We need to show that f(cx + y, z) = cf(x, z) + f(y, z).

Using the definition of f, we have:

f(cx + y, z) = L1(cx + y) L2(z)

Expanding the expression using the linearity of L1, we get:

f(cx + y, z) = (cL1(x) + L1(y)) L2(z)

Using the linearity of L2, we can further simplify the expression:

f(cx + y, z) = cL1(x)L2(z) + L1(y)L2(z)

This can be rewritten as:

f(cx + y, z) = cf(x, z) + f(y, z)

which shows that f satisfies linearity in the first argument.

Linearity in the second argument:
Similarly, let's consider vectors x, y, z in V and a scalar c in F. We need to show that f(x, cy + z) = cf(x, y) + f(x, z).

Using the definition of f, we have:

f(x, cy + z) = L1(x) L2(cy + z)

Expanding the expression using the linearity of L2, we get:

f(x, cy + z) = L1(x)(cL2(y) + L2(z))

Using the linearity of L1, we can further simplify the expression:

f(x, cy + z) = cL1(x)L2(y) + L1(x)L2(z)

This can be rewritten as:

f(x, cy + z) = cf(x, y) + f(x, z)

which shows that f satisfies linearity in the second argument.

Conclusion:
Since f satisfies linearity in both the first and second arguments, it is a bilinear form on V.