In this chapter we study tensor products from a purely algebraic viewpoint. Our approach is to define tensors as functionals that act on bilinear forms. We explain how the tensor product can be seen as a “linearizing space” for bilinear mappings. Tensors can also be viewed as bilinear forms, or as linear mappings and we explore the connections between these ideas. In finite dimensions, tensor products provide a means of understanding the duality of spaces of linear mappings or bilinear forms, either through “tensor duality” or the equivalent “trace duality”. Finally, we look at some examples of tensor products.
KeywordsVector Space Tensor Product Bilinear Form Dual Space Bilinear Mapping
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