Normed vector spaces and Banach spaces

Question

Let $X$ be a Banach space with norm $||.||$ and let $S$ be a non-empty subset in $X$. Let $F_b(S,X)$ be the vector space of $F(S,X)$ of all functions $f:S \rightarrow X$ such that $\{||f(s)||:s \in S\}$ is bounded and consider the norm $||.||_b$ as $F(S,X)$ given by $$||f||_b = \sup \{ ||f(s)|| : s \in S\}.$$

I have to show that $F_b(S,X)$ is a Banach space. Meaning that I have to show that it is a complete metric space. If I let $d(f,g)=||f-g||_b$, I can show that it is a metric space. However, I'm not able to prove that every Cauchy sequence in $F_b(S,X)$ would converge to some element in $F_b(S,X)$.

Answer 1

Let $(f_n)$ be a Cauchy sequence in $F_b(S,X)$.

• For $x \in S$, show that $(f_n(x))$ is a Cauchy sequence in $X$. Therefore, there exists $f(x) \in X$ such that $f_n(x) \to f(x)$.
• Show that $f \in F_b(S,X)$. Hint: $||f(x)|| \leq ||f(x)-f_n(x)||+||f_n(x)||$ for all $n \geq 0$ and $x \in S$.
• Show that $f_n\to f$ in $F_b(S,X)$. Hint: for all $x \in S$ and $n,m \geq 0$, $$||f_n(x)-f(x)|| \leq ||f_n(x)-f_m(x)||+||f_m(x)-f(x)||$$