All the vector spaces I have seen so far in my class have ordinary component wise addition. This means I can add the corresponding components of vectors or matrices or polynomials, etc. But are there cases where the addition is not really the familiar component addition but rather something different than wise addition like function composition or something like that?

  • $\begingroup$ the set of all real continuous functions forms a vector space with the usual addition (f+g)(x)=f(x)+g(x) and scalar multiplication (cf)(x)=cf(x) $\endgroup$ Apr 1 '17 at 5:34
  • $\begingroup$ See this. $\endgroup$
    – Mark Viola
    Apr 1 '17 at 5:36
  • $\begingroup$ @LiChunMin that's still pointwise == coordinatewise addition. $\endgroup$ Apr 1 '17 at 5:53
  • $\begingroup$ All vector space addition is just componentwise addition wrt a basis. This is what bases are for. $\endgroup$ Apr 1 '17 at 5:55
  • $\begingroup$ @Henno Brandsma For the case of (not finite dimensional) vector spaces of functions, you can no longer speak of decomposition on a basis, in the general case. $\endgroup$
    – Jean Marie
    Apr 1 '17 at 6:06

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