• Kaboom@reddthat.com
    link
    fedilink
    English
    arrow-up
    10
    arrow-down
    6
    ·
    6 months ago

    A vector is a list of numbers, at its most basic. You can add a lot of extra functionality to it, but at its core, its just a list.

    • holomorphic@lemmy.world
      link
      fedilink
      English
      arrow-up
      4
      ·
      6 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.

      • Lojcs@lemm.ee
        link
        fedilink
        English
        arrow-up
        1
        arrow-down
        1
        ·
        edit-2
        6 months ago

        It still can be, just not on infinite precision as nothing can with fp.

        • holomorphic@lemmy.world
          link
          fedilink
          English
          arrow-up
          3
          ·
          edit-2
          6 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.