Title: Is Itself an Exponent of the Deadpan Non-Sequitur
Author: #Claudine #Nemejanski
Language: #ASL (#American #Sign #Language), modern (note: represented using illustrations of a posed, #wooden #mannequin)
Depth of recovery: 1,001m
Description: A series of #examples and counter-examples to the proposition that the #proof of the #Banach-#Tarski #paradox both relies [...]

#scp #1986

1/2

One very elementary #Banach space fact is the following:
"A normed space is Banach iff all absolutely convergent series converge".

This comes from the fact that we can find a subsequence (𝑠ₙ) of every Cauchy sequence whose telescopic series
∑(𝑠ₙ₊₁−𝑠ₙ),
the 'series of increments that moves each element 𝑠ₙ to the next 𝑠ₙ₊₁', is absolutely convergent.

And, from absolute convergence we find that the sequence of partial sums is Cauchy.

P.S: Note that absolute *convergence* is always guaranteed from a Cauchy sequence (not necessarily convergent), because we are forming a cauchy sequence ∑‖𝑠ₙ₊₁−𝑠ₙ‖ in the set of real numbers, which is already complete.

King Solomon may just have been misunderstood.

He didn't want to cut the child in half.
Instead he just suggested to Banach-Tarsky it, to help both mothers.
#KingSolomon #BanachTarsky #Banach #Tarsky

'Ridges, Neural Networks, and the Radon Transform', by Michael Unser.

http://jmlr.org/papers/v24/22-0227.html

#neuron #ridges #banach