Some musings on Cauchy-Schwarz, Part III

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In this part we look at the very trivial case of one-dimensional inner product spaces over R. As an exercise you can check any details omitted below, and identify the famous named theorems we are discussing the one-dimensional versions of!

Almost everything discussed below is just the trivial one-dimensional case of standard facts about finite-dimensional, real inner product spaces.

More advanced readers may wish to skip or merely skim Parts III and IV and move on to Part V.

The standard inner product on R is just the usual product in R, so, for x,y∈R,

⟨x,y⟩R=xy.

This is, of course, the one-dimensional version of the dot product!

You can check that this really is a (real) inner product. (This is mostly just the standard rules of arithmetic in R, along with properties of modulus.)

Unsurprisingly, the associated norm is given by

||x||R=√⟨x,x⟩R=√x2=|x|,

i.e., the usual modulus of x.

In this special case, the Cauchy-Schwarz inequality claims that, for all x,y∈R we have 

|xy|≤|x||y|,

but of course we actually have equality here. So the inequality is clearly true here (remembering that equality does imply ≤).

Now let V be any one-dimensional inner product space over R, with inner product ⟨⋅,⋅⟩V and associated norm ‖⋅‖V.

Let {v} be a basis of V (here v can, of course, be any non-zero element of V). Then v≠0, so we normalize by setting

e=1‖v‖Vv.

Then {e} is (trivially) an orthonormal basis of V. (This is effectively the initial step of a Gram-Schmidt process, but it finishes already at this step.)

Define a map T:R→V by T(x)=xe. Then T is easily seen to be a bijection. Moreover, T is a linear isomorphism, and for all x,y∈R, we have

‖T(x)‖V=‖x‖R=|x|

and

⟨T(x),T(y)⟩V=⟨x,y⟩R=xy.

In other words, T preserves norms and inner products.

From this we see that all one-dimensional real inner product spaces are essentially the same as R with its standard inner product (which is just the usual product). Unsurprisingly, this means that equality always holds in the Cauchy-Schwarz inequality in one-dimensional real inner product spaces. You can check this directly, or use the "identification" above.

In the next post we look at the almost equally trivial case of two-dimensional real inner product spaces, and compare these with R2 and C. (Some of this was discussed already in Part II.)

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