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Banachalgebra ABanachspaceX together
with an internal operation, usually called multi-
plication, satisfying the following: for all x,
B y, z ∈ X, α ∈ C

(i.) x(yz) = (xy)z
(ii.) (x +y)z = xz+yz, x(y +z) = xy +xz
(iii.) α(xy) = (αx)y = x(αy)
B¨acklund transformations Transforma-
tions between solutions of differential equations, (iv.) xy ≤ x y
in particular soliton equations. They can be used (v.) X contains a unit element e ∈ X, such
to construct nontrivial solutions from the trivial that xe = ex = x
solution. (vi.) e = 1.
Formally: Two evolution equations u t =
K(x, u, u , ..., u ) and v = G(y,v,v , ..., v ) Banach fixed point theorem Let (X, d) be
m
1
n
1
t
are said to be equivalent under a B¨acklund a complete metric space and T : X → X a
transformation if there exists a transformation contractionmap. ThenT hasauniquefixedpoint
of the form y = ψ(x, u, u , ..., u ), v = φ(x, x ∈ X , i.e., T(x ) = x .
n
1
0
0
0
u, u , ..., u ).
n
1
Banach manifold A manifold modeled on a
Banach space.
bag An unordered collection of elements,
including duplicates, each of which satisfies
Banachspace Anormedvectorspacewhich
some property. An enumerated bag is delimited
is complete in the metric defined by its norm.
by braces ({x}).
Comment: Unfortunately, the same delimiters
barrel A subset of a topological vector space
areusedforsetsandbags. Seealsolist, sequence,
which is absorbing, balanced, convex, and
set, and tuple.
closed.
Baire space A space which is not a count- barreled space A topological vector space
able union of nowhere dense subsets. Example: E is called barreled if each barrel in E is a neigh-
A complete metric space is a Baire space. borhood of 0 ∈ E, i.e., the barrels form a neigh-
borhood base at 0.
Baker-Campell-Hausdorff formula For
barycenter The barycenter (center of mass)
any n × n matrices A, B we have
of the simplex σ = (a , ..., a ) is the point
0 p
s 2 1
−sA
sA
e Be = B +s[A, B]+ [A, [A, B]]+··· b = (a + ··· + a ).
2 σ p + 1 0 p
balanced set A subset M of a vector space V barycentric coordinates Let p , ..., p be
0
n
over R or C such that αx ∈ M, whenever x ∈ M n + 1 points in n-dimensional Euclidean space
n
and |α|≤ 1. E that are not in the same hyperplane. Then for
n
each point x ∈ E there is exactly one set of real
numbers (λ , ..., λ ) such that
n
0
ball Let (X, d) be a metric space.An open
ball B (x ) of radius a about x is the set of all
a 0 0 x = λ p + λ p + ··· + λ p
1 1
0 0
n n
x ∈ X such that d(x, x )<a. The closed ball
0
B (x ) ={x ∈ X | d(x, x ) ≤ a}. and
¯
0
0
a
λ + λ + ··· + λ = 1.
n
0
1
Banach Stefan Banach (1892–1945). Pol- The numbers (λ , ..., λ ) are called barycentric
0 n
ish algebraist, analyst, and topologist. coordinates of the point x.
© 2003 by CRC Press LLC