Banach Space -- from Wolfram MathWorld
|
|
|
Created, developed, and nurtured by Eric Weisstein at Wolfram Research
|
Banach Space
A Banach space is a complete vector space
and
hold for all In the finite-dimensional case, all norms are equivalent. An infinite-dimensional space can have many different norms.
A basic example is
is a Banach space, where
On the other hand, the set of continuous functions on the unit interval
is not a Banach space because it is not complete. For instance, the Cauchy sequence of functions
does not converge to a continuous function. Hilbert spaces with their norm given by the inner product are examples of Banach spaces. While a Hilbert space is always a Banach space, the converse need not hold. Therefore, it is possible for a Banach space not to have a norm given by an inner product. For instance, the supremum norm cannot be given by an inner product. Renteln and Dundes (2005) give the following (bad) mathematical joke about Banach spaces: Q: What's yellow, linear, normed, and complete? A: A Bananach space.
SEE ALSO: Besov Space, Complete Space, Hilbert Space, Minimal
Banach Space, Prime Banach Space, Reflexive
Space, Schauder Fixed Point Theorem,
Vector Space
Portions of this entry contributed by Mohammad Sal Moslehian Portions of this entry contributed by Todd Rowland Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34, 2005. Moslehian, Mohammad Sal; Rowland, Todd; and Weisstein, Eric W. "Banach Space." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BanachSpace.html Wolfram Web Resources
|
with a norm 
. Two
norms 
and 
are called equivalent if they give the same 
and 
such that
.
-dimensional 
given by
denotes the 
with the norm
of a function 
Contact the MathWorld Team
inner product