Author here. It's a good point, and I updated that snippet of text to account for it.
But seriously, you don't think that saying things like "a nasty abomination of a product space," and "they’re only related in that their underlying sets of vectors are built from pairs of vectors in V and W" is enough to drive that point home?
I do struggle in balancing the amount of overly deep/technical bits on my blog with the easier to understand mathematics. If only my audience was primarily HN! :)
To me, "a nasty abomination of a product space" reads like "a product space which is a nasty abomination". The "only related ..." bit is good, but it comes much later in the article; it's the early bits that I think are liable to confuse.
I do appreciate, by the way, that explaining nontrivial mathematics to anything other than an audience of very good mathematicians is really bloody hard!
That's a good point. Maybe the word "adulteration" or "bastardization" would be better? Or maybe to avoid odd connotations I should just say it in long form: taking a mathematical machete to a product space and making it nearly unrecognizable.
I do still think, despite what many have said, that a comparison and distinction needs to be made between usual products and tensor products. I think "modifying a product space" happens to be a different way of thinking about it (maybe not the best way), and for what it's worth it helps me keep track of the damn things.
But seriously, you don't think that saying things like "a nasty abomination of a product space," and "they’re only related in that their underlying sets of vectors are built from pairs of vectors in V and W" is enough to drive that point home?
I do struggle in balancing the amount of overly deep/technical bits on my blog with the easier to understand mathematics. If only my audience was primarily HN! :)