If is linear then if and only if which is the (equivalent) conclusion that some authors write instead of It follows that if is also , meaning that holds for all then if and only Every norm is a seminorm and both are symmetric balanced sublinear functions. A sublinear function is a seminorm if and only if it is a balanced function. On a real vector space (although not on a complex vector space), a sublinear function is a seminorm if and only if it is symmetric. The identity function on is an example of a sublinear function that is not a seminorm.
if is linear then if and only if which is the (equivalent) conclusion that some authors write instead of it follows that if is also  meaning that holds for all then if and only every norm is a seminorm and both are symmetric balanced sublinear functions a sublinear function is a seminorm if and only if it is a balanced function on a real vector space (although not on a complex vector space) a sublinear function is a seminorm if and only if it is symmetric the identity function on is an example of a sublinear function that is not a seminorm