Is the map from l^infinite to L(l^2,l^2) a bijection?

[Funky1981]

Homework Statement

Let L(l^2,l^2) be the space of bounded linear operators K:l^2->l^2.

Now I define a map from l^infinite to L(l^2,l^2) as a->Ta(ei) to be Ta(ei)=aiei where ei is the orthonormal basic of l^2 and a=(a1,a2,...) is in l^infinte

I want to prove this map is bijection
can anyone give me some helps??

2. The attempt at a solution
I finished the part of injective but I don.t know how to show it is surjective. I tried to construct a sequence fn s.t. T(fn)=f where f(en)=fn then to show fn is indeed in l^infinite. But I failed to find such sequence

 

[micromass]

I would try to find a counterexample.