Does the weak law of large numbers hold in a general Banach space? Let $B$ be a Banach space. Let $X_n$ be i.i.d. random variables on $B$ with expectation $\mu$. 
Is it true that their empirical mean converges to $\mu$ in law? If not, are there additional assumptions? 
A reference would also be welcome. 
 A: This follows from the scalar LLN. Let $X\in L^1(\Omega, B)$ be a random variable, with $\Omega$ being the probability space. Since $L_1(\Omega)\otimes B$ is dense in $L^1(\omega,B)$, for every $\epsilon>0$ there exist scalar random variables $Y_1,\dots,Y_N\in L^1(\Omega)$ and unit vectors $v_1,\dots,v_N\in B$ such that 
$$
E\left(\left\|X - \sum  Y_k v_k\right\|\right) < \epsilon^2/2
$$
If $S_{n,k}$ is the average of $n$ independent instances of $Y_k$, then by the scalar LLN,
$$
P(\|S_{n,k} - E(Y_k)\|>\epsilon/(2N))<\epsilon 
$$
for all sufficiently large $n$. Hence, 
$$
P\left(\left\|S_{n} - E(X)\right\|>\epsilon \right) \le 
P\left(\left\|S_{n} - \sum S_{n,k}v_k\right\|> \epsilon/2\right) 
+ P\left(\left\|\sum S_{n,k}v_k - E(X)\right\|> \epsilon/2\right)  
\le (2/\epsilon) E\left(\left\|X - \sum  Y_k v_k\right\|\right)
+ \max_k P(\|S_{n,k} - E(Y_k)\|>\epsilon/(2N)) \\ 
< \epsilon +  \epsilon = 2\epsilon
$$
Page 190 of the aforementioned book Probability in Banach spaces by Ledoux and Talagrand  is relevant here, although the result is not exactly stated there for general Banach spaces.
