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).
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