I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ...(adsbygoogle = window.adsbygoogle || []).push({});

I need some help with a further aspect of the proof of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ...

The text of Theorem 6.10 reads as follows:

The above proof mentions Figure 6.1 which is provided below ... as follows:

In the above text, in the proof of Theorem 6.10 under "PROOF OF EXISTENCE" we read:

" ... ... The bilinearity of [itex]b[/itex] shows that [itex]B_1[/itex] maps [itex]V_0[/itex] to [itex]0[/itex]. By Proposition 2.25, [itex]B_1[/itex] descends to a linear map [itex]B \ : \ V_1/V_0 \longrightarrow U[/itex], and we have [itex]Bi = b[/itex]. "

My questions are as follows:

Question 1

Can someone please give a detailed demonstration of how the bilinearity of [itex]b[/itex] shows that [itex]B_1[/itex] maps [itex]V_0[/itex] to [itex]0[/itex]?

Question 2

Can someone please explain what is meant by "[itex]B_1[/itex] descends to a linear map [itex]B \ : \ V_1/V_0 \longrightarrow U[/itex]" and show why this is the case ... also showing why/how [itex]Bi = b[/itex] ... ... ?

Hope someone can help ...

Peter

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*** EDIT ***

The above post mentions Proposition 2.25 so I am providing the text ... as follows:

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*** EDIT 2 ***

After a little reflection it appears that the answer to my Question 2 above should "fall out" or result from matching the situation in Theorem 6.10 to that in Proposition 2.25 ... also I have noticed a remark of Knapp's following the statement of Proposition 2.25 which reads as follows:

So that explains the language: "[itex]B_1[/itex] descends to a linear map [itex]B \ : \ V_1/V_0 \longrightarrow U[/itex]" ... ...

BUT ... I remain perplexed over question 1 ...

Peter

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# I Tensor Product - Knapp - Theorem 6.10 ... Further Question

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