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

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

I need help in order to fully understand the proof of Theorem 12.9 on pages 271-272 ... ...

The relevant text (Theorem 12.8 together with the preceding definition, Definition 12.8) reads as follows:

In the above text from Browder, at the start of the proof of Theorem 12.9, we read the following:

" ... ... It is immediate from the definition of the tensor product that

\(\displaystyle \tilde{u}^{j_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{j_r} ( u_{k_1} ..... u_{k_r} ) = \delta^{ j_1 \cdot \cdot \cdot j_r }_{ k_1 \cdot \cdot \cdot k_r } \)

... ... ... "

Can someone please demonstrate how/why

\(\displaystyle \tilde{u}^{j_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{j_r} ( u_{k_1} ..... u_{k_r} ) = \delta^{ j_1 \cdot \cdot \cdot j_r }_{ k_1 \cdot \cdot \cdot k_r } \)

is true ... ... ?

(Please note that I am slightly lost and overwhelmed by the indices in this theorem ... )

Hope someone can help ...

Peter

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

I am uncertain of the nature of the \(\displaystyle \tilde{u}^{j_i}\) ... but a basis with similar notation was given in the proof of Theorem 12.2 ... so I am providing the text of Chapter 12 up to and including Theorem 12.9 to give readers the context and notation of the Chapter ... ... and any necessary preliminary definitions and results ... as follows:

Hope that helps ...

Peter

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

I need help in order to fully understand the proof of Theorem 12.9 on pages 271-272 ... ...

The relevant text (Theorem 12.8 together with the preceding definition, Definition 12.8) reads as follows:

In the above text from Browder, at the start of the proof of Theorem 12.9, we read the following:

" ... ... It is immediate from the definition of the tensor product that

\(\displaystyle \tilde{u}^{j_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{j_r} ( u_{k_1} ..... u_{k_r} ) = \delta^{ j_1 \cdot \cdot \cdot j_r }_{ k_1 \cdot \cdot \cdot k_r } \)

... ... ... "

Can someone please demonstrate how/why

\(\displaystyle \tilde{u}^{j_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{j_r} ( u_{k_1} ..... u_{k_r} ) = \delta^{ j_1 \cdot \cdot \cdot j_r }_{ k_1 \cdot \cdot \cdot k_r } \)

is true ... ... ?

(Please note that I am slightly lost and overwhelmed by the indices in this theorem ... )

Hope someone can help ...

Peter

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

I am uncertain of the nature of the \(\displaystyle \tilde{u}^{j_i}\) ... but a basis with similar notation was given in the proof of Theorem 12.2 ... so I am providing the text of Chapter 12 up to and including Theorem 12.9 to give readers the context and notation of the Chapter ... ... and any necessary preliminary definitions and results ... as follows:

Hope that helps ...

Peter

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