An inner product on a vector space V is a function ⟨·,·⟩: V × V → ℝ satisfying positivity (⟨v,v⟩ ≥ 0), symmetry, and linearity in the second argument. The dot product is the standard inner product on R^n. Inner products induce norms and enable defining orthogonality and projections.
You already know the dot product on ℝⁿ: multiply corresponding components and add. What you may not have considered is that the dot product is just one example of a broader structure. An inner product is an abstract operation ⟨·, ·⟩ on a vector space that captures the essential features of the dot product through three axioms: it must be symmetric (⟨u, v⟩ = ⟨v, u⟩), linear in each argument, and positive definite (⟨v, v⟩ ≥ 0, with equality only when v = 0). Any operation satisfying these axioms is an inner product, and the resulting structure is called an inner product space.
The payoff of abstracting the dot product is that geometric concepts — length, angle, and orthogonality — transfer to spaces that have no visual geometry. Given an inner product, you can define the norm (length) of a vector as ‖v‖ = √⟨v, v⟩, and the angle between two vectors via cos θ = ⟨u, v⟩/(‖u‖ ‖v‖). Two vectors are orthogonal when ⟨u, v⟩ = 0. These definitions reproduce familiar geometry in ℝⁿ and generalize it to function spaces, polynomial spaces, and any other vector space where an appropriate inner product exists.
A concrete non-dot-product example: on the space of continuous functions on [0, 1], define ⟨f, g⟩ = ∫₀¹ f(x)g(x) dx. This satisfies all three axioms. Two functions are orthogonal in this space when their product integrates to zero over [0, 1]. The sine and cosine functions that appear in Fourier series are orthogonal in exactly this sense — Fourier series is essentially decomposing a function into orthogonal components, which is the same idea as decomposing a vector in ℝⁿ into its projections onto orthogonal basis vectors.
Positive definiteness is the most subtle axiom and the one most worth understanding deeply. It ensures that ⟨v, v⟩ = 0 forces v = 0 — no nonzero vector has zero "length". Without this, the induced norm would not be a genuine norm, and the analogy with geometric length would collapse. The other axioms (symmetry and linearity) ensure the inner product behaves algebraically like multiplication; positive definiteness ensures the geometric interpretation holds.