  We study Banach envelopes for commutative symmetric sequence or function
spaces, and noncommutative symmetric spaces of measurable operators. We
characterize the class $(HC)$ of quasi-normed symmetric sequence or function
spaces $E$ for which their Banach envelopes $\widehat{E}$ are also symmetric
spaces. The class of symmetric spaces satisfying $(HC)$ contains but is not
limited to order continuous spaces. Let $\mathcal{M}$ be a non-atomic,
semifinite von Neumann algebra with a faithful, normal, $\sigma$-finite trace
$\tau$ and $E$ be as symmetric function space on $[0,\tau(1))$ or symmetric
sequence space. We compute Banach envelope norms on $E(\mathcal{M},\tau)$ and
$C_E$ for any quasi-normed symmetric space $E$. Then we show under assumption
that $E\in (HC)$ that the Banach envelope $\widehat{E(\mathcal{M},\tau)}$ of
$E(\mathcal{M},\tau)$ is equal to $\widehat{E}(\mathcal{M},\tau)$
isometrically. We also prove the analogous result for unitary matrix spaces
$C_E$.
