Some musings on Cauchy-Schwarz: Part I

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I'm currently thinking about the Cauchy-Schwarz inequality for multiple reasons, including the following:
  • I'm currently teaching a Level 4 module on Linear Analysis which includes material on inner product spaces and Hilbert spaces;
  • I'm also still the Teaching Support Officer in our school, and one of the first-year students asked me why the dot product formula in terms of components gives the same answer as the formula in terms of norms and angles, as in
    a⋅b=||a||⋅||b||cosθ, where θ is the angle between a and b.
For the first-year student, I drew a picture to illustrate the special case in R2  where a has the form (||a||,0) and b is written in the form (||b||cosθ,||b||sinθ). I then noted that the two formulae clearly agreed here, and so, as long as you know that the dot product formulae are invariant under rotations about the origin, the general fact follows. 

Now I'm not always interested in the quickest and most elegant proof of a result, if there is another reason why the result must be true. So while teaching material on the Cauchy-Schwarz inequality at level 4 (in a module taken by students in years 3 and 4), I've been thinking again about the underlying whys and wherefores. Since there are plenty of short and elegant proofs of the Cauchy-Schwarz inequality, I don't usually go this far back. However here are some thoughts that I plan to explore in the following parts.

  • Cauchy-Schwarz in complex inner product spaces can be deduced from Cauchy-Schwarz in real inner product spaces. This is essentially because every complex inner product space can also be regarded as a real inner product space with the same norm, but where the real inner product is the real part of the complex inner product. The last bit of the deduction uses a fairly standard "rotation trick": for any complex number z, we have
    |z|=max{Re(αz):α∈C,|α|=1}.
  • The general versions of (real or complex) Cauchy-Schwarz follow from the versions in dimensions ≤2, because there are always only two vectors involved, and their span is an inner product space of dimension ≤2.
  • Cauchy-Schwarz in R2 is an immediate consequence of the "one-dimensional Cauchy-Schwarz for C", which just says that |ˉzw|≤|z||w|. (Of course we actually have equality here.) Moreover, this gives another approach to the equality a⋅b=||a||⋅||b||cosθ briefly discussed above.
  • Which came first, the chicken ... er, the Cauchy-Schwarz inequality or the triangle inequality? In many of the infinite-dimensional examples of inner product spaces in the Linear Analysis module (e.g. ℓ2) you need to verify that you really have a vector space (mainly checking that the space is closed under addition), and that the inner product is well-defined. Of course, once you know you have an inner product space, the Cauchy-Schwarz inequality is automatically true. But typically you need at least some "feeble versions" of the triangle inequality and the Cauchy-Schwarz inequality in order to check that you have an inner product space in the first place.
  • Schwarz and Schwartz are different mathematicians!

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