MTG 6256, Fall 2004: Non-book Problem 1
(corrected 8/27/04)
Definition. For p∈R3, let Fp={f|fis a smooth real-valued function defined on some open
neighborhood of p}. Define an equivalence relation ∼on Fpby declaring f∼gif there exists
an open neighborhood of pon which fand gcoincide. An equivalence class under this relation
is called a smooth germ, and the set Gp:= Fp/∼of equivalence classes is called the space of
smooth germs of functions at p. Note that all elements of a given equivalence class have the
same value at p, so if ˆ
f∈ Gp, the number ˆ
f(p) is well-defined.
(a) Give an explicit necessary and sufficient condition for a function f∈ Fpto represent
the 0-germ, the equivalence class of the constant function 0. Do the same for the equivalence
class of a general constant function (a constant germ).
(b) Show that the usual operations of addition of functions and multiplication by scalars
induce well-defined operations on germs, and therefore that Gpis a vector space, whose zero
element is the 0-germ, under these operations.
(c) Show that multiplication of functions induces a well-defined operation on germs. (We
still call this operation multiplication, and denote it the same way we do mulitiplication of
functions.)
Definition. A linear functional L:Gp→Ris called Leibnizian (or Leibniz-linear) if for all
ˆ
f, ˆg∈ Gpwe have L(ˆ
fˆg) = L(ˆ
f)ˆg(p) + ˆ
f(p)L(ˆg). (“Functional” is another word for real-valued
function that is often used when the domain is some infinite-dimensional object.)
(d) Show that if L:Gp→Ris Leibnizian then L(any constant germ) = 0.
Let Lp={Leibnizian functionals Gp→R}.
(e) Show that Lpis a vector space (with the obvious operations and zero element).
(f) Show that Lpis naturally isomorphic to Tp(R3). (Hint: Taylor’s Theorem [not Taylor
series]. You have to know the right version of Taylor’s Theorem for this hint to be useful; the
one most commonly taught in modern Calculus 1-2-3 courses, while pretty, is useless for many
purposes, including this one.)
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