Some musings on Cauchy-Schwarz, Part Va

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In this part, we take as our starting point the (easy) results discussed so far for real inner product spaces of dimensions less than or equal to $2$. That is, that for a given dimension $n$ (and we only need $n\leq 2$ below), up to "real inner product space isomorphism", all $n$-dimensional inner product spaces over $\R$ are the same as $\R^n$ with the usual dot product. In particular, the Cauchy-Schwarz inequality holds in all real inner product spaces of dimension $\leq 2$, with equality if and only if the two vectors involved are linearly dependent.

Now let $V$ be any inner product space over $\R$, with inner product $\langle \cdot,\cdot \rangle_V$ and associated norm $\|\cdot\|_V$. (Here $V$ can be infinite-dimensional, and need not be separable.)

Let $\vx, \vy \in V$, let $W$ be the linear subspace of $V$ spanned by $\vx$ and $\vy$, i.e., $\lin\{\vx,\vy\}$. Clearly $W$ is then itself a real inner product space of dimension at most $2$. Thus the Cauchy-Schwarz inequality holds for $\vx$ and $\vy$ in $W$, and hence also in $V$, with equality holding if and only if the set $\{\vx,\vy\}$ is linearly independent. 

This shows that the Cauchy-Schwarz inequality for general real inner product spaces follows from the results in dimensions less than or equal to $2$. 

In Part Vb we will recall how the result for complex inner product spaces follows from the real inner product space version.

Notice that the earlier discussion gives us a rather strange root through this bit of theory.

  • Cauchy-Schwarz is (of course) completely trivial for $0$-dimensional inner product spaces.
  • Cauchy-Schwarz holds, with equality, for all $1$-dimensional inner product spaces over $\R$ or $\C$.
  • Cauchy-Schwarz for $2$-dimensional real inner product spaces follows from Cauchy-Schwarz for the $1$-dimensional complex inner product space $\C$.
  • Cauchy-Schwarz for all real inner product spaces follows from Cauchy-Schwarz for real inner product spaces of dimension less than or equal to $2$.
  • Cauchy-Schwarz for complex inner product spaces follows from the result for real inner product spaces.
There is nothing surprising here, but perhaps (?) some of this helps to see why the Cauchy-Schwarz inequality just has to be generally true, as an alternative to simply proving (the real case) true using e.g. a short and elegant proof involving quadratics and discriminants. On the other hand, you may well feel that the proof using quadratics and discriminants is the real reason that the result is true. But maybe this series of elementary posts gives an alternative viewpoint that might make some people (e.g. me) just that little bit happier?

I may go back and have another look at the Hahn-Banach Theorem in real dimensions less than or equal to $2$ or $3$ to see if this gives me a similar way to understand why the argument needed in the  Hahn-Banach Theorem to extend by one real dimension just has to work. (In the past I had decided that it works "for Hahn-Banach reasons".)

Whether I will ever find a similar way to think about the Cohen Factorization Theorem is another matter!

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