Question

If f is a function in function space with a zero norm, then would it be necessary to have for f a totally zero function?

Answer 1

Yes, this follows from the axioms of norms. Namely, one axiom is that if the norm of a vector (which can be a function in a function space) is zero, then the vector should be the zero vector, i.e. the zero function.

Answer 2

Or at least it should be zero with the usual function norms (for example in \(L^p\) space). It would be possible to come up with other norms in vector spaces that define some non-zero function to be the zero vector, but usually this is not the case.

Answer 3

You would have to find a really weird addition for the function space to get a additive 0 that is not the zero function though.

Answer 4

If f is a function in function space and \[\left[ f,f \right]\] = 0, then f(x) need not be identically zero for all x, but it can differ from zero only on
a set of measure zero.

Answer 5

Yup that's true.

Answer 6

All depends on what function space we are talking about.

Answer 7

Yes it is necessary, This comes from the axiom norms That if the norm of a vector is zero,
then the vector should be the zero vector