Originally Posted by

**Bongo** Hello,

Here are two Algebras and i want to show that they are isomorphic:

1)$\displaystyle

\mathcal{L}(\bigotimes_{i\in \mathbb{N}} (E_i)) \

$

2)$\displaystyle \bigotimes_{i\in \mathbb{N}} \mathcal{L}(E_i)$

The algebra of the form $\displaystyle \mathcal{L} (E)$ is the set of all endomorphisms. And the $\displaystyle E_i$ are vector spaces.

I try to construct a isomorphism, but it failed all the time. Can you please help me?

I think that the Algebra-Multiplikation is composition in both cases.

I have defined this map, but i couldn't show that it is an isomorphism:

$\displaystyle \phi: \bigotimes_{i\in \mathbb{N}} \mathcal{L}(E_i) -> \bigotimes_{i\in \mathbb{N}} \mathcal{L}(E_i) , \phi (\otimes f_i):=f $ with f defined as: $\displaystyle f: \bigotimes E_i -> \bigotimes E_i, \; f(\otimes e_i):= \otimes f_i(e_i)$

Regards