Without Loss of Generality in Differential Equations
- Suppose we want to prove a theorem [Zyg, vol.1, p.59, Lemma 8.11] for the
general case. In order to restrict our consideration to a standard case without loss of
generality, we invent a device X which allows us to transform the general case to a standard case.
Example. Suppose fÎL1.
By [Zyg, vol.1, p.51, l.-2-l.-1], S~[f] at x0
is the same thing as S~[y]
at t = 0, where y(t) = [f(x0+t)-f(x0-t)]/2.
Thus, X(f) = y is the device that allows us to assume
that x0 = 0 and that f is odd without loss
of generality. One may wonder whether the device X can produce a new element if
x0 = 0 and f is odd. The answer is no: X(f) = f
[Zyg, vol.1, p.59, l.-9]. Consequently, X(X(f)) = X(f).
Before we prove the theorem for the standard case, we should
verify that X(f) satisfies the hypothesis of the theorem just like f does. After
we prove the theorem for the standard case, we should be able to derive the
conclusion of the general case from the conclusion of the standard case.