Even, odd

I very much like doing the repn theory of even/odd functions:
the $C_2$ action on $\cF(\bR,\bR)$ via $f(x) \mapsto f(-x)$,
and how it direct sum decomposes (into even/odd subspaces),
and the projections $\half (f(x)+f(-x))$ etc.

...and how this is the same as $(x,y) \to (y,x)$
(with basis $(1,1), (1,-1)$)
...but you could also look at the basis $(1,0),(0,1)$,
which get swapped -- of course many do, but these are -orthogonal-.

...and in fact the even/odd functions break up as a sum of these
(plus a trivial one at $0$)
...and this is baby repn theory.

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Further, this is the same setup you get with $y''=y$
(which is 2 dim'l), hence $e^x,e^{-x}$, cosh and sinh
...and also the $y''=-y$, and $e^{ix}, e^{-ix}$ and cos/sin
(note that here we have an action of $C_4$, not $C_2$;
 if you want to be -very- correct, then we're looking at
 $y''''=y$, which breaks up as $y''=y$ and $y''=-y$
 (by looking at the $C_2 < C_4$ action))

...and you get the same picture!
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Also, note that this action preserves the functional $f \mapsto \int f$,
and thus also the norm $\norm{f} \mapsto \int \norm{f}$
and the corresponding (Hermitian) inner product $<f,g> = \int \bar f g$
...and thus the direct sum is orthogonal w/r/t this inner product:
odd functions are perpendicular to even functions.
[This is particularly relevant for orthogonal polynomials
 on a symmetric interval, like [-1,1]]
