metric space, complete

Question

Let $(Y,\tau)$ be a topological space, with $Y\neq 0$ and let $(X,d)$ be a metric space. It is $C_b(Y,X):=\{f:Y\to X|\quad\text{f ist bounded and continuous}\,\}$

$D(f,g)=sup_{y\in Y} d(f(y),g(y))$ is the metric on $C_b(Y,X)$

Task:

Show, that $C_b(Y,X)$ is complete, iff $X$ is complete.

Hello,

I have a question to this task.

"$\Rightarrow$"

Let $C_b(Y,X)$ be complete. Let $n,m\in\mathbb{N}$, then

$d(x_n,x_m)\leq sup_{y\in Y} d(x_n,x_m)=D(x_n,x_m)<\epsilon$.

Since $C_b(Y,X)$ is complete, it exists a $N\in\mathbb{N}$ such that $D(x_n,x_m)<\epsilon$ for every $m,n>N$.

"$\Leftarrow$"

Let $X$ be a complete.

$D(x_n, x_m)=sup_{y\in Y} d(x_n,x_m)<sup_{y\in Y} \epsilon=\epsilon$.

Since $X$ is complete for every $\epsilon>0$ exists an $N\in\mathbb{N}$, such that $d(x_n,x_m)<\epsilon$ for every $m,n>N$.

Is this proof correct? Thanks in advance for every help.

Answer 1

Your method is true but the elements to choose and analyze the cauchy property and convergence are a little inappropriate. I give my proof only for the $\Rightarrow$. The other side is the similar. Also the following needs some details that you can refine. Furthermore, there is a similar proof in the Real Analysis of "Aliprantis".

Let $ C_{b}(Y,X) $ is a complete metric space. And let $ \{f_{m}(y)\}_{m\in \mathbb{N}}$ is cauchy sequence in $X.$ So there exist $m,n \in \mathbb{N}$ such that $d(f_{m}(y),f_{n}(y))<\epsilon$.

Since $ C_{b}(Y,X) $ is a complete metric space and, $D(f_{m},f_{n})=\sup_{y\in Y}d(f_{m}(y),f_{n}(y))<\epsilon $ then, there exists $f\in C_{b}(Y,X)$ such that $f_{m}\to f $ this gives us $D(f_{m},f)<\epsilon\, \Rightarrow\, d(f_{m}(y),f(y))<\epsilon$ so $f(y)\in X $

Answer 2

I find your notation (in your proof) incomprehensible. For the first part,the following may be what you have in mind:

Suppose $d$ is an incomplete metric. Let $(x_n)_n$ be a $d$-Cauchy sequence with no limit point.

Let $f_n(y)=x_n$ for all $y\in Y.$ Then $D(f_m,f_n)=d(x_m,x_n)$ so $(f_n)_n$ is a $D$-Cauchy sequence.

Consider any $f\in C_b(Y,X)$ and any $y\in Y.$ There exists $m\in N$ such that $f(y)\not\in Cl_X\{x_n:n\geq m\},$ otherwise $f(y)\in \cap_{m\in N}Cl_X\{x_n:n\geq m\}=\emptyset.$

So there exists $m\in N$ and $r>0$ and such that $r=\inf_{n\geq m}d(f(y),x_n)=\inf_{n\geq m}d(f(y),f_n(y))\leq \inf_{n\geq m}D(f,f_n).$

So $(f_n)_n$ does not converge to to any $f\in C_b(Y,X)$ and so $C_b(Y,X)$ is incomplete.