0

Following this related Example of tensor (0,2) acting on two vectors (Nick's answer). I want to ask why in the definition we consider the map $(f\otimes g)\colon V\otimes W \to X\otimes Y$ and not $(f\otimes g)\colon V\times W \to X\otimes Y$ as $(f\otimes g)(a,b)=f(a)\otimes g(b)$ ?

Why we need $V\otimes W$ here ? A lot of times I see the notation $(f\otimes g)(a,b)$ as well. Is this some kind of convention we make if we work over fields just because we always have $F\otimes F\cong F$ ?

Maybe there is a lot to unpack here but a reference to how we define the tensor product action on vectors and covectors would also be nice.

領域展開
  • 2,631
  • A bilinear map is not a linear map. We consider $V \otimes W$ because it is the linear map induced, by the universal property, from the map $V \times W \to X \otimes Y$. – user480840 Feb 14 '24 at 20:01

2 Answers2

1

Start with that $h \; V \times W \to X \otimes Y$ as $h(a,b) = f(a) \otimes g(b)$.

Take $a = c_1 a_1 + c_2 a_2$. Then $h(a,b) = c_1 h(a_1,b) + c_2 h(a_2,b)$. Similarly for $b$ linear combinations.

Now use universal property. This way we stay within the realm of talking about vector spaces and linear maps instead of talking about a mix of linear, bilinear etc.

AHusain
  • 5,241
0

You can consider the map $f * g\colon V \times W \to X \otimes Y$, $(a, b) \mapsto f(a) \otimes g(b)$ (choosing different notation to distinguish the maps, but this is not standard) without issue, but this map is bilinear and not linear. Thus, by the universal property of the tensor product, it induces a linear map $f \otimes g\colon V \otimes W \to X \otimes Y$, $a \otimes b \mapsto f(a) \otimes g(b)$. It's often nicer to have this rather than the former since most other maps you're considering are usually linear and you have all these tools for understanding linear maps at your disposal, but $f * g$ and $f \otimes g$ are closely related and "almost the same" (take this with a grain of salt), and therefore one sometimes doesn't distinguish too rigorously.

Ben Steffan
  • 8,325