What Vol I left on the table
Volume I treated the dot product as a formula: u . v = u_1 v_1 + ... + u_n v_n. From it you read off the norm ||v|| = sqrt(v . v), the angle between vectors, and orthogonality. But a formula tied to one basis hides what really matters. Volume II asks: which properties made the dot product useful, and can we keep just those?
An inner product on a real vector space V is a map <u, v> taking two vectors to a scalar, obeying three axioms: it is symmetric (<u, v> = <v, u>), linear in the first argument, and positive-definite (<v, v> > 0 for v != 0). Every fact you loved about the dot product follows from these three lines alone.
Why complex scalars need a conjugate
Over C the naive copy breaks. If we kept <v, v> = sum v_k^2 with v = (i, 0), we'd get i^2 = -1 < 0 — a negative length. The fix is to conjugate one side: <u, v> = sum u_k * conj(v_k). Now <v, v> = sum |v_k|^2 >= 0 again. This gives the complex (Hermitian) inner product.
The price: the product is no longer symmetric but conjugate-symmetric, <u, v> = conj(<v, u>), and it is linear in the first slot but conjugate-linear in the second: <u, c*v> = conj(c) <u, v>. A form linear in one argument and conjugate-linear in the other is a sesquilinear form — Latin for "one and a half" linear.
Inner products beyond R^n
The abstraction pays off because the axioms hold on spaces that are not column vectors at all. On the space of continuous functions on [a, b], <f, g> = integral of f(x) g(x) dx is a genuine inner product — and it is exactly the engine behind Fourier series. On matrices, <A, B> = trace(A^T B) measures "overlap" entry by entry.
A weighted inner product on R^2 (still valid!):
<u, v> = 2*u1*v1 + 5*u2*v2
Check the axioms with the Gram matrix G = [2, 0; 0, 5]:
<u, v> = u^T G v
symmetric: G = G^T yes
positive-definite: 2 > 0 and 5 > 0 yes (eigenvalues of G positive)
Same vectors, different geometry: the "unit circle"
{ v : <v,v> = 1 } is now an ellipse 2*x^2 + 5*y^2 = 1.That last example is general: given any basis, the numbers G_ij = <e_i, e_j> form the Gram matrix, and <u, v> = u-coords^T G v-coords. An inner product on a finite-dimensional space is the same data as a symmetric (Hermitian over C) positive-definite matrix. The dot product is just the case G = I.