\mathbb{F}_{q^m} $-linear subspace $ D = D_1 \oplus D_2 \oplus \cdots \oplus D_l \subseteq C $, which satisfies $ \dim(D) = r $. By definition, it holds that $ D^* = D_1^* + D_2^* + \cdots + D_l^* $. Hence $$\begin{split}
{\rm wt_R}(D) & \leq {\rm wt_R}(D_1) + {\rm wt_R}(D_2) + \cdots + {\rm wt_R}(D_l) \\
 & = d_{R,r_1}(C_1^\prime) + d_{R,r_2}(C_2^\prime) + \cdots + d_{R,r_l}(C_l^\prime),
\end{split}$$ and the result follows again by Lemma \[equivalent definition slides\].

Observe that the bound (\[upper bound\]) is valid with the same proof for a general $ \mathbb{F}_{q^m} $-linear code that can be decomposed as a direct sum of $ \mathbb{F}_{q^m} $-linear subcodes $ C = C_1^\prime \oplus C_2^\prime \oplus \cdots \oplus C_l^\prime $.

\[remark exact grw product\] In the general setting of Theorem \[grw bounds\], the same result as in Corollary \[exact grw product\] holds whenever $ C_i $ and $ C_i^\prime $ are rank equivalent (see Section \[sec all optimal\]), for each $ i = 1,2, \ldots, l $, since in that case it holds that $ d_{R,r}(C_i) = d_{R,r}(C_i^\prime) $ for all $ i =1,2, \ldots, l $ and all $ r = 1,2, \ldots, k_i $.

MRD reducible codes with MRD 