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Given a map P: V-->W for vector spaces V and W and the map P*: W* --> V* we have the relationship that many of us are familiar with:

For e in V, f in W, E in V* and F in W*, we can say that

(P*(F))(e)=F(P(e)).

This is nice and fine. So this is kind of the case for a rank 1 tensor. Now can anyone help me generalize this to rank r tensors, namely those of type (0,r) and (r,0)? We can't worry about the case of type (r,s) unless we know that P is an invertible map. But I'm having a REALLY REALLY hard time understanding the case for some higher rank tensor.

I am also hoping that whoever can help with this can introduce it with also explaining which of these induced maps is the pushforward and which is the pullback.

Thanks.

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# Induced maps on tensor spaces

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