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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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