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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 Rn. 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.