Metric Spaces MT23, Function spaces
Flashcards
Suppose $X$ is a metric space. Can you define $B(X)$?
\[B(X) = \{f : f(X) \text{ is bounded}\}\]
where each $f : X \to \mathbb R$.
Suppose $X$ is a metric space and that $f \in B(X)$. Can you define $ \vert \vert f \vert \vert _ \infty$?
\[\sup _ {x \in X} \vert f(x) \vert \]
Suppose $X$ is a metric space. Can you define $C(X)$?
\[C(X) = \{f : f \text{ is continuous}\}\]
where each $f : X \to \mathbb R$.
Suppose $X$ is a metric space. Can you define $C _ b(X)$?
\[C(X) \cap B(X)\]
where each $f : X \to \mathbb R$.
Proofs
Prove that if $X$ is a metric space, then $ \vert \vert f \vert \vert _ \infty$ defines a valid norm.
Todo.