Thanks! I am in Maine for the weekend, doing math.
Rn is a Banach space, where you measure the length of a vector in the usual way, by taking the square root of the dot product of a vector with itself. If that's the only kind of vector space you ever think about, then the concept of a Banach space seems kind of boring. The advantage of the definition is that you can come up with either weirder vectors or weirder concepts of length, and then use the Banach space axioms to transfer all of your intuition about the way length in vector spaces work to this new, stranger context.
Two fundamental examples of different norms are the taxicab norm, |(x1,...,xn)| = |x1|+...+|xn|, and the max norm, |(x1,...,xn)| = max{|x_1|, ..., |x_n|}. The source of the name of the taxicab norm is that if you're trying to get from place to place in a city, you have to follow the rectangular street grid, rather than taking diagonal shortcuts. One way I think about the max norm is trying to find a way to encapsulate people's tastes mathematically. If Alisaundra and Bailey rate a bunch of movies and Alisaundra picks all 7s but Bailey picks all 8s, they probably have a lot in common, despite the slight difference in scores. Then if Cassandra comes along and rates everything between 7 and 8 except Phantom Menace, which she scores at 0, she is much more likely to fight with Alisaundra and Bailey about movies, because the greatest difference is the one that predicts disagreement.
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Banach spaces part 1: different norms
Date: 2019-05-11 03:54 pm (UTC)Rn is a Banach space, where you measure the length of a vector in the usual way, by taking the square root of the dot product of a vector with itself. If that's the only kind of vector space you ever think about, then the concept of a Banach space seems kind of boring. The advantage of the definition is that you can come up with either weirder vectors or weirder concepts of length, and then use the Banach space axioms to transfer all of your intuition about the way length in vector spaces work to this new, stranger context.
Two fundamental examples of different norms are the taxicab norm, |(x1,...,xn)| = |x1|+...+|xn|, and the max norm, |(x1,...,xn)| = max{|x_1|, ..., |x_n|}. The source of the name of the taxicab norm is that if you're trying to get from place to place in a city, you have to follow the rectangular street grid, rather than taking diagonal shortcuts. One way I think about the max norm is trying to find a way to encapsulate people's tastes mathematically. If Alisaundra and Bailey rate a bunch of movies and Alisaundra picks all 7s but Bailey picks all 8s, they probably have a lot in common, despite the slight difference in scores. Then if Cassandra comes along and rates everything between 7 and 8 except Phantom Menace, which she scores at 0, she is much more likely to fight with Alisaundra and Bailey about movies, because the greatest difference is the one that predicts disagreement.