inner product
An inner product is the device that lets a vector space remember angles. The familiar dot product u · v = u_1 v_1 + u_2 v_2 + ... is the prototype: it computes lengths through <v, v> = |v|^2 and detects perpendicularity through u · v = 0. An abstract inner product carries this same power — lengths, angles, projection — to spaces of functions and sequences.
Over the real numbers an inner product is a symmetric bilinear form <·, ·> that is positive definite: <x, x> >= 0 with equality only when x = 0. Over the complex numbers it is sesquilinear — linear in one argument, conjugate-linear in the other — and conjugate-symmetric, <x, y> = conjugate(<y, x>), again with <x, x> > 0 for x nonzero. Either way it induces the norm ||x|| = sqrt(<x, x>), and the angle between vectors is read off from <x, y> = ||x|| ||y|| cos(theta).
The single most important consequence is the Cauchy–Schwarz inequality |<x, y>| <= ||x|| ||y||, which guarantees the formula for the angle makes sense (the cosine stays in [-1, 1]) and from which the triangle inequality for the induced norm follows. Two vectors with <x, y> = 0 are called orthogonal, and orthogonality is the geometric engine of Hilbert space theory.
On l^2 the inner product is <x, y> = sum over n of x_n conjugate(y_n). For x = (1, 1/2, 1/4, ...) and y = (1, -1/2, 1/4, ...), <x, y> = 1 - 1/4 + 1/16 - ... = 1/(1 + 1/4) = 4/5, a convergent series because Cauchy–Schwarz bounds it by ||x|| ||y|| < infinity.
The l^2 inner product of two square-summable sequences, convergent by Cauchy–Schwarz.
The complex conjugate in the sesquilinear case is not a quibble: it is what keeps <x, x> real and nonnegative, so that ||x|| = sqrt(<x, x>) is well-defined. Conventions differ on which slot is linear; physics texts usually make the second argument linear, mathematics texts the first.