Let . We have . Now you can use the definition of continuity.
I need the solution of the following problem:
Show that in a normed space , vector addition and multiplication by scalars are continuous operations with respect to the norm; that is, the mappings defined by and are continuous.
This problem in "Introductory Functional Analysis with applications, Erwin Kreyszig" problem 4 page 70.
Be sure that you can show that is true.
We show that is continuous.
Suppose that is a scalar and is a point in the normed space.
We start the proof the way we start each continuity proof.
Suppose that . Now let
If we have and then