We first show that the functions

are linearly independent for any two distinct values of

p.

Consider the fourier transform formulae shown on p.166. The first formula expresses

as an arbitrary linear combination of the functions

, where

gives the "coefficient" or "weight" of each function in the (integral) combination.

If the functions

were NOT linearly independent, then there would be two or more distinct hyperfunctions

that would give rise to exactly the same

. (By the definition of linear independence).

However, each of the fourier equations given is the inverse of the other. So there is a bijection between the space of hyperfunctions

and the space of hyperfunctions

. Or to put this another way, one can use the 1st formula to compute

given an initial

, and the 2nd formula to recover the initial

from

.

This would be impossible if there were more than one distinct

that corresponded to a particular

. So the functions

are indeed linearly independent for any two distinct values of

p.

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The fourier formulae of p.166 may readily be extended to higher dimensions, and then proof that the functions

are linearly independent for any two distinct one-forms

follows analogously. (We could then take

so as to correspond exactly to the expressions in the book, if we wished to).