Function Spaces

Abstract

Points converge to a limit if they get physically closer and closer to it. What about a sequence of functions? When do functions converge to a limit function? What should it mean that they get closer and closer to a limit function? The simplest idea is that a sequence of functions f _n converges to a limit function f if for each x, the values f _n (x) converge to f (x) as n → ∞. This is called pointwise convergence: a sequence of functions f _n : [a, b] → ℝ converges pointwise to a limit function f : [a, b] → ℝ if for each x ∈ [a, b],

$$ \mathop {\lim }\limits_{n \to \infty } {\mkern 1mu} {\mkern 1mu} {f_n}\left( x \right){\mkern 1mu} = {\mkern 1mu} f\left( x \right). $$

The function f is the pointwise limit of the sequence (f _n ) and we write

$$ {f_n}{\mkern 1mu} \to {\mkern 1mu} f{\mkern 1mu} {\text{or}}{\mkern 1mu} \mathop {\lim }\limits_{n \to \infty } {\mkern 1mu} {\mkern 1mu} {f_n}{\mkern 1mu} = {\mkern 1mu} f. $$

Note that the limit refers to n → ∞, not to x → ∞. The same definition applies to functions from one metric space to another.