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- Why is the tensor product of two tensors again a tensor?

> **Exercise.** Let T1and T2be tensors of type (r1 s1)and (r2 s2) respectively on a vector space V. Show that T1⊗

T2can be viewed as an (r1+r2 s1+s2)tensor, so that the

> tensor product of two tensors is again a tensor, justifying the

> nomenclature...

What I’m reading：《An introduction to tensors and group theory for physicists》Authors: Jeevanjee, Nadir. According to it，

a tensor is a multilinear function that eats r vectors as well as s dual vectors and produces a number...And "Give two finite-dimensional vector spaces V and W,we define their tensor product V⊗W to be the set of all C-valued bilinear functions on V*×W *..."

How do I define the tensor product of two tensors by these definition?And What the target object this tensor product acts on should look like？

T2can be viewed as an (r1+r2 s1+s2)tensor, so that the

> tensor product of two tensors is again a tensor, justifying the

> nomenclature...

What I’m reading：《An introduction to tensors and group theory for physicists》Authors: Jeevanjee, Nadir. According to it，

a tensor is a multilinear function that eats r vectors as well as s dual vectors and produces a number...And "Give two finite-dimensional vector spaces V and W,we define their tensor product V⊗W to be the set of all C-valued bilinear functions on V*×W *..."

How do I define the tensor product of two tensors by these definition?And What the target object this tensor product acts on should look like？