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.

 

[tacman]

Yes, we do need to justify the fact that linearity is still applied even on infinite sums. This is because linearity is a fundamental property of linear functionals, and it must hold for all possible inputs, including infinite sums. In order to justify this, we can use the linearity property of the functional f, which states that for any scalars a and b and any two sequences x and y in L1, we have f(ax+by) = af(x) + bf(y).

In this case, we can view the sequence x as an infinite sum of the basis sequences (E1, E2, ...). Using the linearity property, we can write x = A1*E1 + A2*E2 + ... = A1*x1 + A2*x2 + ..., where x1, x2, ... are the basis sequences. Then, we can apply the linearity property again to write f(x) = f(A1*x1 + A2*x2 + ...) = A1*f(x1) + A2*f(x2) + ...

Therefore, we can justify that linearity is still applied on infinite sums by using the linearity property of the functional and breaking down the infinite sum into a finite sum of basis sequences.