Comment on Listen here, Little Dicky

• iAvicenna@lemmy.world ⁨3⁩ ⁨days⁩ ago

  Doesn’t BCT imply that an infinite dimensional Banach spaces cannot have a countable basis

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  • gandalf_der_12te@discuss.tchncs.de ⁨2⁩ ⁨days⁩ ago

    Uhm, yeah, but there’s two different definitions of basis iirc. And i’m using the analytical definition here; you’re talking about the algebraic definition.

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    • 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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