Function Spaces

A function space is a vector space where "vectors" are functions


  • functions are not just $n$ points like vectors, but they are the entire continuum
  • this is a vector space, but "vectors" are functions: they have inner product with $\int$ instead of $\sum$, and they also have orthogonality
  • Inner Product: $\langle f, g \rangle = \int\limits_{-\infty}^{\infty} f(x) \, g(x) \, dx$
  • Orthogonal Functions: if the inner product of $f$ and $g$ is 0 then $f$ and $g$ are orthogonal
  • this is an inner product for functions: we multiply the values for every $x$ and sum them using integral