V* of a vector space
V over ℝ matches lists of reals to linear functionals.
What’s the simplest way to say this? Talk about a number like “5”. Initially I think of it as 5 stones ⬤⬤⬤⬤⬤. But I could also imagine a line through the origin with a slope of 5, representing the verb
pictures of lines through the origin with various slopes
Seen as a function ƒ₅=quintuple, the-line-through-the-origin-with-a-slope-of-5, is
ƒ₅(x)=5•x. That ƒ₅ does things like
- ƒ₅(■■■)=5+5+5 and
Counting in the dual space ƒ₀,ƒ₁,ƒ₂,… would look like
_ / ∕ ...|. Increasing slope from _ to ⁄ to | instead of increasing number from 0 to 1 to ∞. Or I could say
(Why did I jump so suddenly 0,1,… from _ flat to ⁄ 45°? This just proves that half of the ℝ⁺ are stuffed between [0,1) and the other half are between (1,∞).
To jump between the two worlds you use the reciprocal map flip(■)≝1/■. T
hen you’d be counting id, half, third, fourth, fifth, sixth, seventh… Infinity in a teacup.)
These two things—the five rocks ⬤⬤⬤⬤⬤ and the function ƒ₅—aren’t even the same kind of thing. One is nouns and one is a verb.
But still, for any real number that I “counted” I could match up a function, just like I did with
- "5 that I counted" and
- "function ƒ₅ =
This one-dimensional story can be upgraded to a multi-dimensional one where
- lists of reals
(3.1, √2, −2.1852, ..., 6)
- many-to-one functions
ƒ( list ) = 3.1•list[first] + √2•list[second] − 2.1852•list[third] + ... + 6•list[Nth].
Translating between the noun and verb viewpoints is then called musical isomorphism, represented with ♭ and ♯ symbols. Raising and lowering indices in a tensor is ♯ and ♭.