Inner Product Spaces

Inner product axioms, the induced norm, and Cauchy–Schwarz.

The idea

An inner product space is a vector space equipped with one extra operation that supplies the geometry — length, angle, perpendicularity — that the vector space axioms alone do not provide.

The model is the dot product on $\mathbb{R}^{n}$: multiply matching coordinates and add, $u_{1}v_{1} + \cdots + u_{n}v_{n}$. That number is large for vectors pointing the same way, zero for perpendicular ones, and applied to a vector with itself it gives the square of the length. The figure shows the configuration every such statement is about: two vectors $u$ and $v$ drawn from a common point, with the angle $\theta$ between them that the inner product makes measurable. The general definition keeps exactly the properties responsible for this behavior.

An inner product on a real vector space $V$ assigns a number $\langle u, v \rangle$ to each pair of vectors, subject to three axioms: it is positive, meaning $\langle v, v \rangle > 0$ whenever $v \neq 0$; symmetric, meaning $\langle u, v \rangle = \langle v, u \rangle$; and linear in its first slot.

Length is then defined rather than assumed. Positivity makes $\langle v, v \rangle$ nonnegative, so its square root exists, and the norm $\lVert v \rVert = \sqrt{\langle v, v \rangle}$ is the length of $v$. Every statement about length, angle or perpendicularity in the space is derived from the three axioms and nothing else.

Ways to work on it

• Walkthrough. The inner product axioms and the norm they induce.
• Practice. Compute an inner product or a norm on ℝ^n.
• Hardest. Test Cauchy–Schwarz and characterize its equality case.

Not sure where to start? Take the ten-question placement test.