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Linear Algebra, Optimisation and Numerical Methods / Inner Product Spaces and Orthogonality

Inner Products, Norms, and the General Cauchy-Schwarz Inequality

The dot product on Rn\mathbb{R}^n (Definition 1 of the LA-I dot-product note) is one specific formula. This note asks what makes that formula work — which properties of v⋅w\mathbf{v}\cdot\mathbf{w} are doing the real work — and defines an inner product as any function with those properties, on any vector space, real or complex. The reward is immediate: every geometric fact LA-I proved for Rn\mathbb{R}^n (Cauchy-Schwarz, the triangle inequality, orthogonality) is re-derived here once, for every inner product space simultaneously, including complex vector spaces and function spaces that have no "arrow" picture at all.
Learning objectives. By the end of this note you should be able to:
  • state the five defining properties of an inner product and verify a candidate function against them;
  • compute the induced norm and recognise when a formula is not norm-inducing;
  • prove the Cauchy-Schwarz inequality for a general inner product space;
  • give at least one inner product space that is not Rn\mathbb{R}^n with the dot product.

In this lesson

  1. The definition
  2. The induced norm
  3. The general Cauchy-Schwarz inequality

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