Let $X_1,X_2$ be disjoint (except for the $0$ vector) Banach subspaces of a topological vector space $X$, i.e.: $$ X_1\cap X_2 = \{0\}. $$ Define a norms on the space $ X_1 + X_2 $ by $$ \|f\|_A\triangleq \max\{\|f_1\|_{X_1},\|f_2\|_{X_2}\}. $$ $$ \|f\|_B\triangleq \inf\{\|f_1\|_{X_1}+\|f_2\|_{X_2}:\, f=f_1+f_2, f_1 \in X_1,f_2 \in f_2\}. $$ Are these two norms equivalent on $X_1+X_2$?
It seems to me to be the case, since all norms are equivalent on $\mathbb{R}^d$ and the disjointnes condition $X_1\cap X_2 =\{0\}$ essentially means that $\|\cdot\|_A$ factors through $\ell^{\infty}_n$ ie: $$ \|f\|_A = \|\|f_1\|_{X_1},\|f_2\|_{X_2}\|_{\ell^{\infty}}... $$ But maybe I'm missing something?...