Overview

Linear Algebra I built the dot product (the-dot-product-length-and-angle) as a fixed formula on $\mathbb{R}^n$: multiply components, add. This topic generalises that single formula into an entire class of objects — inner products — defined by the properties they must satisfy rather than any one recipe, opening the door to complex vector spaces, function spaces, and every other setting where "angle" and "length" are meaningful concepts. This is the same axiomatic move Linear Algebra I made for vector spaces themselves (vector-spaces-and-subspaces), now applied one level up, to the geometric structure sitting on top of a vector space.