11 - Function Spaces

Bounded Space of Functions

If $X$ is any set then $B(X)$ is the space of functions

$$f:X\to \mathbb{R}\text{ such that }f(X)=\{f(x):x\in X\}\text{ is bounded}$$

If $f\in B(X)$ then define

$$||f|{|}_{\infty }=\underset{x\in X}{\sup }|f(x)|$$

Vector Space and Norm of a Bounded Space of Functions lemma

For any set $X$ then

$$B(X)\text{ is a vector space}\text{ and }||\cdot |{|}_{\infty }\text{ is a norm}$$

Space of Continious Functions

Let $X$ be a metric space then

$$C(X)\text{ is the space of all continuous functions }f:X\to \mathbb{R}$$

Space of Continious Function is a Vector Space lemma

Space $C(X)$ a vector space over $\mathbb{R}$ with pointwise addition and scalar multiplication

Space of Continuous Bounded Functions

Let $X$ be a metric space then
$C_B$ is the space of continuous, bounded functions on $X$ defined as

$$C_b(X):=C(X)\cap B(X)$$

Since $C_b()X)$ is a subspace of $B(X)$ then it inherits $||f|{|}_{\infty }={\sup }_{x\in X}|f(x)|$

Hence we can define the metric $d_{\infty }$ on $C_{b}X)$ with

$$d_{\infty }(f,g):=||f-g|{|}_{\infty }$$