How to Conquer Tensorphobia

If you really want to impress your friends and confound your enemies, you can invoke tensor products… People run in terror from the symbol. He was explaining some aspects of multidimensional Fourier transforms, but this comment is only half in jest; people get confused by tensor products. People who really understand tensors feel obligated to explain it using abstract language (specifically, universal properties). And the people who explain it in elementary terms don't really understand tensors. This post is an attempt to bridge the gap between the elementary and advanced understandings of tensors. We'll start with the elementary (axiomatic) approach, just to get a good feel for the objects we're working with and their essential properties.