Function Spaces (𝑋 → 𝑌 or 𝑌^𝑋)

Function Spaces - Examples

Function Spaces that are also Vector Spaces

• a finite-dimensional vector space (𝑉,𝐹) where 𝑉 is a set of all polynomial functions of degree 𝑘 or less
  • example finite basis #1 {1, 𝑥, 𝑥², …, 𝑥^𝑘} # “standard basis”
  • example finite basis #2 {4, 6𝑥, 1𝑥², …, 2𝑥^𝑘}
• an infinite-dimensional vector space (𝑉,𝐹) where 𝑉 is a set of all polynomial functions of any degree
  • example infinite basis #1 {1, 𝑥, 𝑥², … } # “standard basis”
  • example infinite basis #2 {4, 6𝑥, 1𝑥², … }
• an infinite-dimensional vector space (𝑉,𝐹) where 𝑉 is a set of functions, where each function (𝑓) outputs a field (𝐹) (i.e. 𝑓: 𝑋 → 𝐹)
  • example infinite basis #1 {1, 𝑥, 𝑥², …, 𝑐𝑜𝑠(𝑥), …, 5^𝑥, … }
• the vector space of all sequences (𝐹ℕ) where it is the set of all functions 𝑓: ℕ → 𝐹
  • any sequence space (i.e. linear subspace of 𝐹^ℕ)
• any 𝐿𝑝 spaces (function space)

Function Spaces that are NOT Vector Spaces

• If the domain of the functions is not a vector space, then the function space is not a vector space

  Let:

  • 𝑋 = {red, green, blue}
  • 𝑌 = {marcus, erina}
  • $f(x)=\{\begin{array}{ll}marcus& \text{if }x=redorgreen\\ erina& \text{if }x=blue\end{array}$

  The domain 𝑋 is not a vector space

Subpages

• Basis Functions
• Calculating Number of Distinct Functions in a Function Space