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Listen here, Little Dicky (mander.xyz)

submitted 1 week ago by [email protected] to c/[email protected]

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[–] [email protected] 1 points 1 week ago* (last edited 1 week ago) (1 children)

So I call an infinite dimensional vector space of countable/uncountable dimensions if it has a countable and uncountable basis. What is the analytical definition? Or do you mean basis in the sense of topology?

[–] [email protected] 2 points 1 week ago (2 children)

Uhm, i remember there's two definitions for basis.

The basis in linear algebra says that you can compose every vector v as a finite sum v = sum over i from 1 to N of a_i * v_i, where a_i are arbitrary coefficients

The basis in analysis says that you can compose every vector v as an infinite sum v = sum over i from 1 to infinity of a_i * v_i. So that makes a convergent series. It requires that a topology is defined on the vector space fist, so convergence becomes well-defined. We call such a vector space of countably infinite dimension if such a basis (v_1, v_2, ...) exists that every vector v can be represented as a convergent series.

[–] [email protected] 2 points 1 week ago (1 children)

Ah that makes sense, regular definition of basis is not much of use in infinite dimension anyways as far as I recall. Wonder if differentiability is required for what you said since polynomials on compact domains (probably required for uniform convergence or sth) would also work for cont functions I think.

[–] [email protected] 1 points 1 week ago

regular definition of basis is not much of use in infinite dimension anyways as far as I recall.

yeah, that's exactly why we have an alternative definition for that :D

Wonder if differentiability is required for what you said since polynomials on compact domains (probably required for uniform convergence or sth) would also work for cont functions I think.

Differentiability is not required; what is required is a topology, i.e. a definition of convergence to make sure the infinite series are well-defined.

[–] [email protected] 2 points 1 week ago

i just checked and there's official names for it:

the term Hamel basis refers to basis in linear algebra
the term Schauder basis is used to refer to the basis in analysis sense.