holomorphic

holomorphic , 2 months ago

Functions from the reals to the reals are an example of a vector space with elements which can not be represented as a list of numbers.

holomorphic , 2 months ago (edited 2 months ago)

But the vector space of (all) real functions is a completely different beast from the space of computable functions on finite-precision numbers. If you restrict the equality of these functions to their extension,

defined as f = g iff forall x\in R: f(x)=g(x),

then that vector space appears to be not only finite dimensional, but in fact finite.
Otherwise you probably get a countably infinite dimensional vector space indexed by lambda terms (or whatever formalism you prefer.)
But nothing like the space which contains vectors like

F_{x_0}(x) := (1 if x = x_0; 0 otherwise)

where x_0 is uncomputable.

holomorphic , 3 months ago

It may have nothing to do with categorization, but has everything to do with categorification which is much more interresting anyway.