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Algebra Multiplikation

  • Thread starter Canavar
  • Start date
  • #1
16
0
Hello,

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

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

How the multiplication is defined on this space?

Regards
 

Answers and Replies

  • #2
354
0
But for this i need to know the algebra-structure on [tex]\bigotimes_{i=1}^\infty \; \L (V_i)[/tex].

How the multiplication is defined on this space?
Multiplication is defined component-wise.
 
  • #3
16
0
Ok, thank you!

Do you know, how i can construct a isomorphism?

Is this a Isomorphism:

[tex]\phi: \bigotimes_{i=1}^\infty L(V_i) \rightarrow L(\bigotimes_{i=1}^\infty V_i)[/tex], defined by [tex]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))[/tex]

i.e. each elm. is send to the product of the [tex]f_i(e_i)[/tex]

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?

Regrads
 
Last edited:
  • #4
16
0
Can nobody help me?:-(
It would be very nice, if someone can help me.

Thanks in advance.
 
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