Some musings on Cauchy-Schwarz, Part IVa

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In this very short part we look briefly at one-dimensional complex inner product spaces, essentially repeating some of the material from previous posts.

As we noted in Part II, the standard (linear in the second variable) complex inner product on C is given by ⟨z,w⟩C=ˉzw, with the associated norm given by the usual modulus, ‖z‖C=|z|.

Since we have (by standard properties of C) that

|⟨z,w⟩C|=|ˉzw|=|z||w|=‖z‖C‖w‖C,

the Cauchy-Schwarz inequality holds for C, and in fact equality always holds here.

We can now repeat the reasoning from Part III to see that all one-dimensional complex inner product spaces are essentially the same as C with its standard inner product (given above). Unsurprisingly, this means that equality always holds in the Cauchy-Schwarz inequality in one-dimensional complex inner product spaces.

In part IVb we will look at two-dimensional real inner product spaces, which are all essentially identical to R2 with the standard dot product. They are also all essentially identical to C when C is regarded as a real inner product space, using the real inner product (as discussed in Part II).

⟨z,w⟩=Re(ˉzw)=Re(z)Re(w)+Im(z)Im(w),

and the Cauchy-Schwarz inequality for all two-dimensional real inner product spaces is immediate from this, because |Re(ˉzw)|≤|ˉzw|. (Of course, Cauchy-Schwarz in R2 is easy to prove anyway!)

Note that equality holds for Cauchy-Schwarz in the real inner product space C if and only if ˉzw lies on the real axis, which happens if and only if 0,z,w are colinear (as expected).

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