Is a norm on a set a continuous function with respect to the topology induced by the norm?
Is a topology on the set that can make the norm continuous (i.e. the topology that is compatible with the norm) not unique? Is it a superset of the unique topology induced by the norm?
I am asking this question, because I heard (I am also not sure if it is correct) that a topology that can make an inner product continuous is not unique (such a topology is called weak topology on the inner product space?), and is a superset of the topology induced by the inner product.
Thanks and regards! Pointers to some references are appreciated!