Definition1
A complete normed vector space is called a
Banach space .
-
with
norm, i.e.,
is a Banach space.
-
for
and
is a Banach space.
-
for
is a Banach space.
- Any finite-dimensional normed vector space is Banach, e.g.,
or
with any norm.
-
with
norm for
is not Banach.
Definition 2
A complete inner product space is called a
Hilbert space .
-
is a Hilbert space.
-
is a Hilbert space.
- Any finite-dimensional inner product space is a Hilbert space.
Note that every Hilbert space is Banach, but the converse is not true.
Hilbert spaces will be extremely important in this course.
