A seminorm is a function 
on a real vector space
or complex vector space 
, denoted 
, such that the following
conditions hold for all 
and 
in 
and any scalar 
.
1. 
,
2. 
, and
3. 
.
Note that it is possible for 
for nonzero 
. For example, the functional 
for continuous
functions on 
is a seminorm which is not a norm. A seminorm is a norm iff 
is equivalent to 
.
