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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 12: Multilinear Algebra ... ...

I am uncertain regarding a formal and rigorous proof that the tensor product is associative ... so I will give my attempt at a proof and then I hope that someone will kindly critique the proof for me ...

I will use the definitions and notation of Sections 12.7 and 12.8 ...

Sections 12.7 and 12.8 read as follows:

We have to show that \(\displaystyle ( \alpha \otimes \beta ) \otimes \gamma = \alpha \otimes ( \beta \otimes \gamma) \)

====================================================================================

Let \(\displaystyle \alpha\) be such that \(\displaystyle \alpha : V^r \to \mathbb{R}\)

Let \(\displaystyle \beta\) be such that \(\displaystyle \beta : V^s \to \mathbb{R}\)

Let \(\displaystyle \gamma\) be such that \(\displaystyle \gamma : V^t \to \mathbb{R} \)

Then we have ...

\(\displaystyle ( \alpha \otimes \beta ) \otimes \gamma (v_1 \ ... \ ... \ v_{ r + s + t } ) \)

\(\displaystyle = ( \alpha \otimes \beta ) (v_1 \ ... \ ... \ v_{ r + s } ) \gamma (v_{ r + s + 1} \ ... \ ... \ v_{ r + s + t } )\)

\(\displaystyle = \alpha (v_1 \ ... \ ... \ v_r ) \beta (v_{ r + 1 } \ ... \ ... \ v_{ r + s } ) \gamma (v_{ r + s + 1 } \ ... \ ... \ v_{ r + s + t } )\) ... ... ... (1)

... and we also have ...

\(\displaystyle \alpha \otimes ( \beta \otimes \gamma) (v_1 \ ... \ ... \ v_{ r + s + t } )

\)

\(\displaystyle = \alpha (v_1 \ ... \ ... \ v_r ) ( \beta \otimes \gamma) (v_{ r + 1 } \ ... \ ... \ v_{ r + s + t } )\)

\(\displaystyle = \alpha (v_1 \ ... \ ... \ v_r ) \beta (v_{ r + 1 } \ ... \ ... \ v_{ r + s } ) \gamma (v_{ r + s + 1 } \ ... \ ... \ v_{ r + s + t } )\) ... ... ... (2)

Now (1), (2) \(\displaystyle \Longrightarrow\) \(\displaystyle ( \alpha \otimes \beta ) \otimes \gamma = \alpha \otimes ( \beta \otimes \gamma) \)

======================================================================================

NOTE: ... ... in the above proof I calculated/processed the tensor product sign in the parenthesis second ... not first as I believe the notation is supposed to mean ...

I did this because I felt that when expanding a tensor product we needed to keep the form

tensor \(\displaystyle \otimes\) tensor (ordered list of variables) = tensor (ordered list) tensor (ordered list)

and processing the tensor product within the parentheses first did not appear to allow this form to be preserved ...

Peter

I am currently reading Chapter 12: Multilinear Algebra ... ...

I am uncertain regarding a formal and rigorous proof that the tensor product is associative ... so I will give my attempt at a proof and then I hope that someone will kindly critique the proof for me ...

I will use the definitions and notation of Sections 12.7 and 12.8 ...

Sections 12.7 and 12.8 read as follows:

We have to show that \(\displaystyle ( \alpha \otimes \beta ) \otimes \gamma = \alpha \otimes ( \beta \otimes \gamma) \)

====================================================================================

Let \(\displaystyle \alpha\) be such that \(\displaystyle \alpha : V^r \to \mathbb{R}\)

Let \(\displaystyle \beta\) be such that \(\displaystyle \beta : V^s \to \mathbb{R}\)

Let \(\displaystyle \gamma\) be such that \(\displaystyle \gamma : V^t \to \mathbb{R} \)

Then we have ...

\(\displaystyle ( \alpha \otimes \beta ) \otimes \gamma (v_1 \ ... \ ... \ v_{ r + s + t } ) \)

\(\displaystyle = ( \alpha \otimes \beta ) (v_1 \ ... \ ... \ v_{ r + s } ) \gamma (v_{ r + s + 1} \ ... \ ... \ v_{ r + s + t } )\)

\(\displaystyle = \alpha (v_1 \ ... \ ... \ v_r ) \beta (v_{ r + 1 } \ ... \ ... \ v_{ r + s } ) \gamma (v_{ r + s + 1 } \ ... \ ... \ v_{ r + s + t } )\) ... ... ... (1)

... and we also have ...

\(\displaystyle \alpha \otimes ( \beta \otimes \gamma) (v_1 \ ... \ ... \ v_{ r + s + t } )

\)

\(\displaystyle = \alpha (v_1 \ ... \ ... \ v_r ) ( \beta \otimes \gamma) (v_{ r + 1 } \ ... \ ... \ v_{ r + s + t } )\)

\(\displaystyle = \alpha (v_1 \ ... \ ... \ v_r ) \beta (v_{ r + 1 } \ ... \ ... \ v_{ r + s } ) \gamma (v_{ r + s + 1 } \ ... \ ... \ v_{ r + s + t } )\) ... ... ... (2)

Now (1), (2) \(\displaystyle \Longrightarrow\) \(\displaystyle ( \alpha \otimes \beta ) \otimes \gamma = \alpha \otimes ( \beta \otimes \gamma) \)

======================================================================================

NOTE: ... ... in the above proof I calculated/processed the tensor product sign in the parenthesis second ... not first as I believe the notation is supposed to mean ...

I did this because I felt that when expanding a tensor product we needed to keep the form

tensor \(\displaystyle \otimes\) tensor (ordered list of variables) = tensor (ordered list) tensor (ordered list)

and processing the tensor product within the parentheses first did not appear to allow this form to be preserved ...

**Can someone please critique the above proof ...**

Peter

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