I have a question regarding norms I know the definition of a

I have a question regarding norms. I know the definition of a norm and the interpretation of the euclidean norm is intuitive because it conforms to our understanding of distance. But in general, of what use is a norm (without referring to abstract properties of a norm).

More specifically, I came across this norm ||f||=supremum{||f(x)||:||x||=1}. Now, I know what the supremum of a set is, but what does :||x||=1 indicate in this norm. Also, is this called the supremum norm?

Solution

Norm is nothing but a funtion from a vector space(for example R_n) to R+, where R+ is the set of positive real numbers. Basically it assigns some positive length to any vector in the underlying Vector space.

Considering your norm, it is called the Supremum norm. || x || = 1 is the condition under which the supremum is taken, That is || f || is supremum of all the |f(x)| for such x which satisfies || x || = 1.