Comment on Listen here, Little Dicky

• iAvicenna@lemmy.world ⁨2⁩ ⁨days⁩ ago

  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?

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  • gandalf_der_12te@discuss.tchncs.de ⁨2⁩ ⁨days⁩ ago

    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.

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    • iAvicenna@lemmy.world ⁨2⁩ ⁨days⁩ ago

      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.

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      • gandalf_der_12te@discuss.tchncs.de ⁨2⁩ ⁨days⁩ 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.

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    • gandalf_der_12te@discuss.tchncs.de ⁨2⁩ ⁨days⁩ 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.

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