A Banach space is a complete vector space 
with a norm 
.
Two norms 
and 
are called equivalent
if they give the same topology, which is equivalent
to the existence of constants 
and 
such that
 |
(1)
|
and
 |
(2)
|
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 
-dimensional
Euclidean space with the Euclidean norm. Usually,
the notion of Banach space is only used in the infinite dimensional setting, typically
as a vector space of functions. For example, the
set of continuous functions on closed interval
of the real line with the norm of a function 
given by
 |
(3)
|
is a Banach space, where 
denotes the supremum.
On the other hand, the set of continuous functions on the unit interval ![[0,1]](/images/equations/BanachSpace/Inline11.svg)
with the norm of a function 
given by
 |
(4)
|
is not a Banach space because it is not complete. For instance, the Cauchy sequence of functions
 |
(5)
|
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.
