A seminorm is a function on a 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 is a seminorm which is
not a norm. A seminorm is a norm if 
is equivalent to 
.
