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I have recently been studying various texts on differential geometry, and I am quite puzzled that various authors define the notion of a tensor quite differently. I have come across the following defintions:

  1. A tensor is a bilinear mapping from $V \times W$ into the field $K$
  2. A tensor is a bilinear mapping from $V^*\times W^*$ into the field $K$
  3. A tensor is an element of the tensor product $V \otimes W$

Why are they equivalent?

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Related: math.stackexchange.com/q/10282/1242 –  Hans Lundmark Jan 14 '13 at 19:25
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