A Hilbert space is a vector space 
over 
or 
with an inner product 
such that the norm defined
by
 |
turns 
into a complete metric space. An inner
product space need not be complete in this
induced norm (Axler 2026, pp. 214 and 224).
Examples of finite-dimensional Hilbert spaces include
1. Euclidean space 
with 
the dot product of
and 
.
2. The complex vector space 
with 
, where the bar denotes the
complex conjugate.
An infinite-dimensional example is the real L2-space 
with Lebesgue
measure. Its elements are equivalence
classes of square integrable real
functions, identified when equal almost everywhere.
In this case, the inner product is
 |
Taking equivalence classes is necessary for positive definiteness, since a function that is zero almost everywhere has norm zero (Axler 2026, pp. 202 and 224).
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).
