(some) functions are vectors

algebracalculusintuition

intuition

• functions here only refer to those that are continuous on the real line
• functions are vectors of infinite components
  • → dot product are integral $f\cdot g=\int f(t)g(t)dt$
  • in fact, the definition of a vector space is

    what

    • a set whose elements made of a field equipped with additive and scalar multiplicative operations satisfying some axioms related to linearity (or vector space axioms)

    Link to original

• for Fourier Transform:
  • the $sin$ and $cos$ functions are basis functions, each frequency is a basis
    • each $sin$ function is like a $[1,0,1,0,1,...]$ vector, different frequencies means different gap size between $0$ and $1$
      • these are odd functions
    • likewise, each $cos$ function is like a $[0,1,0,1,0,...]$
      • these are even functions
    • we can use Euler’s identity formula to combine them to span the whole function space
  • any function $f$ is a linear combination of them, just like any vector is a linear combination of the basis vectors
  • to know how much a basis vector contribute to a vector $v$, we project $v$ to the basis vector
    • → thus to know how much a basis function (frequency) contribute to $f$, we project $f$ to the basis function

references

• Functions are vectors - YouTube