A Hilbert space is a vector space 
with an inner product 
such that the norm
defined by
 |
turns 
into a complete metric space. If the metric
defined by the norm is not complete,
then 
is instead known as an inner product space.
Examples of finite-dimensional Hilbert spaces include
1. The real numbers 
with 
the vector dot product
of 
and 
.
2. The complex numbers 
with 
the vector dot product
of 
and the complex
conjugate of 
.
An example of an infinite-dimensional Hilbert space is 
, the set of
all functions 
such that the integral
of 
over the whole real
line is finite. In this case, the inner
product is
 |
A Hilbert space is always a Banach space, but the converse need not hold.
A (small) joke told in the hallways of MIT ran, "Do you know Hilbert? No? Then what are you doing in his space?" (S. A. Vaughn, pers. comm., Jul. 31, 2005).
