Space of bounded functions vs. bounded space of functions.

Question

Suppose I have a bounded set of functions, say $B\subset C[0,1]$. What exactly does this mean? I.e. is a bounded set of continuous functions equivalent to a set of continuous bounded functions? For reference, I'm proving an operator $T:X\to Y$ is compact showing that $T(B)\subset Y$ is precompact for every bounded set $B\subset X$.

Answer 1

No, a bounded set of continuous functions is different from a set of bounded continuous functions. In your case of $C[0,1]$, note that every $f \in C[0,1]$ is a bounded continuous function, due to the compactness of $[0,1]$. A subset $B \subseteq C[0,1]$ is a bounded set if there is one bound for all the functions in $B$, that is the number $$ \sup_{f \in B} \|f\|_\infty $$ exists (i. e., is finite).

Answer 2

In the context of your question a set $B$ is bounded means that it is a collection of vectors that are all bounded in norm by some value $M$.

However the norm you are probably using on $C[0,1]$ is $\| x \| = \sup_{t \in [0,1]} |x(t)|$. This should show you the connection between your two phrases.