Inner products let mathematicians measure angles in spaces nobody can picture
School geometry measures length and angle with rulers and protractors. An inner product space captures the same ideas algebraically, through a single operation that takes two vectors and returns a number. That abstraction lets length, angle and perpendicularity make sense even in spaces with infinitely many dimensions, which is where much of modern analysis lives.
The model is the familiar dot product: multiply matching coordinates of two vectors and add the results. An inner product generalises this to any real or complex vector space, provided it obeys a few rules. It must be linear in one argument, it must satisfy a symmetry condition, and the inner product of any nonzero vector with itself must be positive. Two vectors whose inner product is zero count as orthogonal, the abstract version of meeting at a right angle. Giuseppe Peano was the first to use a vector space equipped with such an operation, in 1898.
Complex numbers force a twist. Swapping the order of the two vectors does not return the same value but its complex conjugate, and the operation is linear in one argument while conjugate-linear in the other. For the complex numbers themselves, the inner product multiplies one number by the conjugate of the other; plain multiplication would fail the positivity rule. Complex inner product spaces are sometimes called unitary spaces.
Conventions differ across disciplines. Mathematicians usually make the first argument the linear one, while many physicists and matrix specialists choose the second. Quantum mechanics has its own bra-ket notation, writing inner products with an angled bracket split by a vertical bar. Over the real numbers the whole structure simplifies to a positive-definite symmetric bilinear form.
Every inner product also defines a length, or norm, so each inner product space is a normed space. If that space is complete, meaning no sequences that ought to converge fall through gaps, it is called a Hilbert space. Any inner product space that falls short can be completed into one, much as the rational numbers are filled out by the reals.
Source: Inner product space
