Let be Banach spaces over the field . Let be dense in , be dense in , such that is closed under -linear combinations. Let , such that for all , , and for all , . Then, there exists such that .
First, we define . Let . There exists , such that . Define . We need to check that is well-defined. Let , . Now, let . Then, since is Cauchy, there exists such that implies . Therefore, so converges in . Now, there exists such that implies . Therefore, , by the triangle inequality. So . Hence
Since were arbitrary we see that is well-defined on .
Now, we check that is linear.
Let . Then, , , where . Then, so is additive. The demonstration that , for is quite similar.
Now, we need to show that is bounded. Thus is bounded with .