section_path: Types of limits/In sequences/Function space
section_name: Function space
paragraph_text: Function space
  
  This section deals with the idea of limits of sequences of functions, not to be confused with the idea of limits of functions, discussed below.
  
  The field of functional analysis partly seeks to identify useful notions of convergence on function spaces. For example, consider the space of functions from a generic set E {\displaystyle E} to R {\displaystyle \mathbb {R} }. Given a sequence of functions { f n } n > 0 {\displaystyle \{f_{n}\}_{n>0}} such that each is a function f n: E → R {\displaystyle f_{n}:E\rightarrow \mathbb {R} }, suppose that there exists a function such that for each x ∈ E {\displaystyle x\in E}, f n (x) → f (x) or equivalently lim n → ∞ f n (x) = f (x). {\displaystyle f_{n}(x)\rightarrow f(x){\text{ or equivalently }}\lim _{n\rightarrow \infty }f_{n}(x)=f(x).}
  
  Then the sequence f n {\displaystyle f_{n}} is said to converge pointwise to f {\displaystyle f}. However, such sequences can exhibit unexpected behavior. For example, it is possible to construct a sequence of continuous functions which has a discontinuous pointwise limit.
  
  Another notion of convergence is uniform convergence. The uniform distance between two functions f, g: E → R {\displaystyle f,g:E\rightarrow \mathbb {R} } is the maximum difference between the two functions as the argument x ∈ E {\displaystyle x\in E} is varied. That is, d (f, g) = max x ∈ E | f (x) − g (x) |. {\displaystyle d(f,g)=\max _{x\in E}|f(x)-g(x)|.} Then the sequence f n {\displaystyle f_{n}} is said to uniformly converge or have a uniform limit of f {\displaystyle f} if f n → f {\displaystyle f_{n}\rightarrow f} with respect to this distance. The uniform limit has "nicer" properties than the pointwise limit. For example, the uniform limit of a sequence of continuous functions is continuous.
  
  Many different notions of convergence can be defined on function spaces. This is sometimes dependent on the regularity of the space. Prominent examples of function spaces with some notion of convergence are Lp spaces and Sobolev space.