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## Main Question or Discussion Point

The definition (taken from Robert Gilmore's: Lie groups, Lie algebras, and some of their applications):

We have two vector spaces [itex]V_1[/itex] and [itex]V_2[/itex] with bases [itex]\{e_i\}[/itex] and [itex]\{f_i\}[/itex]. A basis for the direct product space [itex]V_1\otimes V_2[/itex] can be taken as [itex]\{e_i\otimes f_j\}[/itex]. So an element w of this space would look like (summation convention):

[tex]w=A^{ij}e_i\otimes f_j [/tex]

For the direct sum space [itex]V_1\oplus V_2[/itex], we take as basis: [itex]\{e_1,e_2,....;f_1,f_2,...\}[/itex].

\end of stuff from Gilmore

My question:

If we take [itex]V_1[/itex] to be the x-axis, and [itex]V_2[/itex] to be the y-axis, we can say that the tensor product space is the y=x line. Since any element would look like: [itex]w=A\;\;\hat{x}\otimes \hat{y} [/itex]

whereas the direct sum space is spanned by [itex]\{\hat{x},\hat{y}\}[/itex], i.e. it consists of the entire [itex]R^2[/itex].

but it seems weird to me that the tensor product space in this example has a smaller dimension than the direct sum space. Obviously I have a misconception, where is it?

Thanks!

We have two vector spaces [itex]V_1[/itex] and [itex]V_2[/itex] with bases [itex]\{e_i\}[/itex] and [itex]\{f_i\}[/itex]. A basis for the direct product space [itex]V_1\otimes V_2[/itex] can be taken as [itex]\{e_i\otimes f_j\}[/itex]. So an element w of this space would look like (summation convention):

[tex]w=A^{ij}e_i\otimes f_j [/tex]

For the direct sum space [itex]V_1\oplus V_2[/itex], we take as basis: [itex]\{e_1,e_2,....;f_1,f_2,...\}[/itex].

\end of stuff from Gilmore

My question:

If we take [itex]V_1[/itex] to be the x-axis, and [itex]V_2[/itex] to be the y-axis, we can say that the tensor product space is the y=x line. Since any element would look like: [itex]w=A\;\;\hat{x}\otimes \hat{y} [/itex]

whereas the direct sum space is spanned by [itex]\{\hat{x},\hat{y}\}[/itex], i.e. it consists of the entire [itex]R^2[/itex].

but it seems weird to me that the tensor product space in this example has a smaller dimension than the direct sum space. Obviously I have a misconception, where is it?

Thanks!

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