/sci/ - Science & Math

>>11624245
>>>/sci/sqt
>>>/wsr/

>>11614940
actually the best 2hu is jo'on and if you could post a high quality version of that picture that'd be great

>>11213042
>>11213080
>constant 1
Ah, I have time, might as well reply properly.
> Is this really a basis for lp, for every single p in the given interval?
Yup.
The norm is p-th root of the sum, so as we match up the terms, the sum gradually disappears.
>It's just hard to think that a single set can span infinite, strictly different vector spaces, since I was pretty used to seeing vector spaces be entirely uniquely defined just by their bases.
A vector space is entirely determined by its base.
A Hilbert space is a geometric object, but you can still derive most of its behavior from an orthonormal basis, due to the good ole norm of the sum formula.
A Banach space is a completely different beast, and can have some very wild geometry to it.
>>11213100
Remember, a Banach space is a topological/geometrical object, not a purely algebraic one. Purely algebraic intuition doesn't work for it.
That definition says that, as the sum goes to infinity, the distance of the sum from v goes to zero, in the sense specifically of topological convergence.

Could Tao beat Villani in a fist fight?