Sum of compact sets

Question

Let $A,B$ two non-empty compact subsets of a normed space X. How can we prove that the set $S=A+B=\{a+b : a \in A, b \in B\}$ is compact?

Here's my reasoning:
Let $\Omega = \{\Omega_1, \Omega_2,…\}$ be an open cover of $S$. $\Omega$ induces two open cover $X,Y$ respectively of $A,B$, where

$X_i = \{a \in A : a+b \in \Omega_i~~for~some~b\}$
$Y_i = \{b \in B : a+b \in \Omega_i~~for~some~a\}$

in practice $\Omega_i = X_i + Y_i$.

Now my idea is build a finite subcover this way: consider a finite subcover $X_F = \{X_j : j \in J\}$ where $J$ is a finite set of indices. If $Y_F = \{Y_j : j \in J\}$ is a finite subcover of $B$, then we are done as $\{\Omega_j : j \in J\}$ is a subcover of $\Omega$. Otherwise we can keep adding indexes to set $J$ until $Y_F$ becomes a finite subcover of $B$.

This is not too formal but I don't want to make a simple question unreadable, I think the idea should be clear.

Does this works? Is there a better/easier way to get the same result?

Answer 1

Remember $+$ is a function defined as

$$+:\begin{cases} A\times B\to A+B \\ (a,b)\mapsto a+b\end{cases}$$

Since addition is continuous on normed spaces and $A+B$ is the image of the compact set $A\times B$ under this map, the image is compact.

Answer 2

Adam's proof is the cleanest. Here is an alternative proof:

Suppose $x_n \in A+B$ is Cauchy. We have $x_n = a_n+b_n$, with $a_n \in A$ and $b_n \in B$. Since $A,B$ are compact, we have $a_{n_k} \to a \in A$ and $b_{n_k} \to b \in B$ for some subsequence $n_k$. Hence $x_{n_k}=a_{n_k}+b_{n_k} \to a+b \in A+B$, and since $x_n$ is Cauchy, we have $x_n \to a+b$. Hence $A+B$ is complete.

Let $\epsilon>0$ and choose finite ${ 1 \over 2}\epsilon$-nets for $A,B$. That is, some finite collection $a_k,b_k$ such that for any $a \in A,b\in B$ there is some $k,k'$ such that $\|a-a_k\| < { 1 \over 2}\epsilon$, $\|b-b_{k'}\| < { 1 \over 2}\epsilon$. Let $x \in A+B$, then $x=a+b$ for some $a \in A,b\in B$. As above, there is some $k,k'$ such that $\|x-(a_k+b_{k'})\| = \|a+b-(a_k+b_{k'})\| \le \|a-a_k\|+ \|b-b_{k'}\| < \epsilon$. Hence the collection $a_k+b_{k'}$ form a finite $\epsilon$-net for $A+B$. Hence $A+B$ is totally bounded.

Answer 3

$(x_n)$ be any sequence in $A+B$. Then we can write $x_n=a_n+b_n$ where $a_n\in A$ and $b_n\in B$. $(a_n)$ is a sequence in $A$, which is compact. Hence it has a subsequence $(a_{n_k})$ converging to some $a\in A$. $(b_{n_k})$ is a sequence in $B$ which is a compact space, so it also has subsequence $(b_{n_{k_l}})$ converging to $b\in B$. $(a_{n_{k_l}})$ also converges to $a$. Then $(x_{n_{k_l}})$ converges to $a+b$. So every sequence in $A+B$ has a convergent subsequence. Hence, it is compact.