Algebra Multiplikation

Hello,

I want to show that the Algebras $$L(\bigotimes_{i=0}^\infty V_i)\; and\; \bigotimes_{i=1}^\infty \; \L (V_i)$$
are isomorphic!

But for this i need to know the algebra-structure on $$\bigotimes_{i=1}^\infty \; \L (V_i)$$.

How the multiplication is defined on this space?

Regards

But for this i need to know the algebra-structure on $$\bigotimes_{i=1}^\infty \; \L (V_i)$$.

How the multiplication is defined on this space?

Multiplication is defined component-wise.

Ok, thank you!

Do you know, how i can construct a isomorphism?

Is this a Isomorphism:

$$\phi: \bigotimes_{i=1}^\infty L(V_i) \rightarrow L(\bigotimes_{i=1}^\infty V_i)$$, defined by $$x=\bigotimes_{i=1}^\infty (f_i) \to \phi(x): \bigotimes_{i=1}^\infty V_i \to \mathbb{K}, \otimes e_i \to \pi (f_i (e_i))$$

i.e. each elm. is send to the product of the $$f_i(e_i)$$

i couldn't show that this is an isomorphism. Therefore i think it is not one. Have you an idea how i can construct an isomorphism. Perhaps by using the universal property of the tensorprodukt?