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frame A set of n linearly independent vec- free action An action ) : M → M, g ∈
g
tors in the tangent space to an n dimensional G, of a group G on a space M which has no fixed
manifold at a point. points, i.e., ) (x) = x implies g = e.
g
Frechet derivative Let V, W be Banach
spaces, U ⊂ V open. A map F : U → W is free Lagrangian In field theory, a
called Frechet differentiable at the point x ∈ U Lagrangian without interaction of the involved
if it can be approximated by a linear map in the fields. The corresponding theory is called a free
form theory.
F(x + h) = F(x) + DF(x)h + RF(x, h)
function An association of exactly one
where DF(x), called the Frechet derivative of F
object from a set (the range) with each object
atx, isabounded(continuous)linearmapfromV
from another set (the domain). This is equivalent
to W, i.e., DF(x) ∈ L(V, W) and the remainder
to defining the function f as a set, f ⊆ A × B.
RF(x, h) satisfies
For f to be a function, it must be the case that if
RF(x, h)
lim = 0. (x, y) ∈ f and (x, z) ∈ f , then y = z.
h→0 h Comment: This last condition is equivalent to
If the map DF : U → L(V, W) is continuous in saying if a = b, f(a) = f(b). Functions are
x, then F is called continuously differentiable also called mappings and transformations. See
1
or of class C .If F is Frechet differentiable also bijection, biological functions, and relation.
at x then F is Gateaux-Levi differentiable at x
and DF(x)h = DF(x, h).If F is Gateaux-
functional The term functional is usually
Levi differentiable at each point x ∈ U and the
used for functions whose ranges are R or C.
map DF : U → L(V, W) is continuous, then
A functional f on a vector space V is a lin-
F is Frechet differentiable at each x. In finite
m
n
dimensions, i.e., V = R ,W = R , this means ear functional if f(x + y) = f(x) + f(y), and
n
m
F : R → R is Frechet differentiable if all f(αx) = αf (x), for all x, y ∈ V and scalar α.
partial derivatives of F exist and are continuous.
functional analysis The theory of infinite
Frechet space A topolocical vector space
dimensional linear algebra, i.e., the theory of
which is metrizable and complete. Examples are
topological vector spaces (e.g., Hilbert, Banach,
Banach and Hilbert spaces.
or Frechet spaces) and linear operators between
Fredholm alternative If T is a compact them (bounded or unbounded).
operatoronaBanachspace, theneither(I−T) −1
exists or Tx = x has a nonzero solution.
functional derivative Let V be a Banach
∗
Fredholm integral equation The equation space, V its dual space, and F : V → R differ-
entiable at x ∈ V . The functional derivative of F
b δF

f(x) = u(x) + k(x, t)f (t)dt. with respect to x is the unique element δx ∈ V ∗
a (if it exists), such that
Fredholm operator A bounded linear oper-
δF
ator T : V → W between Banach spaces V DF(x) · y =< ,y >, for all y ∈ V
and W such that δx
(i.) Ker(T ) = T −1 (0) is finite dimensional; where <, >: V ×V → R is the pairing between
∗
∗
(ii.) Rang(T ) = T(V ) is closed; V and V and DF is the Frechet derivative of F,
(iii.) W/T (V ) is finite dimensional. i.e., DF(x) · y = lim t→0 t 1 [F(x + ty) − F(x)].
Example: If V = W and K is a compact oper-
ator, then I − K is Fredholm. functional motif A pattern present more
The integer ind T = dim T −1 (0) − dim W/ than once in a set of biochemical reactions,
T(V ) is called the index of T. described from any point of view.

© 2003 by CRC Press LLC