A question on linearity of functionals

[Ricky2357]

Suppose we have a bounded linear functional f defined on L1 (the sequence space of all absolutely summable sequences) and we take the natural (Schauder) basis for L1, that is, the set of sequences (E1,E2,...,En,...) that have 1 in the n th position and everywere else zero. Pick x in L1.
Then x=A1*E1+A2*E2+... , for some scalars An.
Do we need to justify the fact that

f(x)=A1*f(E1)+A2*f(E2)+... ?

In other words, do we need to justify that linearity is still applied even on infinite sums?

 

[matt grime]

There is nothing to justify at all.