Coordinate projections and inclusions on a finite product Banach space

By [L1], $\mathrm{\parallel }(x,y){\mathrm{\parallel }}_{\max }\ge \mathrm{\parallel }x\mathrm{\parallel }$ and $\mathrm{\parallel }(x,y){\mathrm{\parallel }}_{\max }\ge \mathrm{\parallel }y\mathrm{\parallel }$, so $\mathrm{\parallel }{\pi }_X(x,y)\mathrm{\parallel }\le \mathrm{\parallel }(x,y){\mathrm{\parallel }}_{\max }$ and $\mathrm{\parallel }{\pi }_Y(x,y)\mathrm{\parallel }\le \mathrm{\parallel }(x,y){\mathrm{\parallel }}_{\max }$. Thus both projections are bounded with operator norm at most $1$ by [L2] and [L3].

For $x\in X$ and $y\in Y$, [L1] gives $\mathrm{\parallel }{\iota }_X(x){\mathrm{\parallel }}_{\max }=\mathrm{\parallel }(x,0){\mathrm{\parallel }}_{\max }=\mathrm{\parallel }x\mathrm{\parallel }$ and $\mathrm{\parallel }{\iota }_Y(y){\mathrm{\parallel }}_{\max }=\mathrm{\parallel }(0,y){\mathrm{\parallel }}_{\max }=\mathrm{\parallel }y\mathrm{\parallel }$. So $\mathrm{\parallel }{\iota }_X\mathrm{\parallel }\le 1$ and $\mathrm{\parallel }{\iota }_Y\mathrm{\parallel }\le 1$, with equality whenever the relevant domain is nonzero. If $X\mathrm{\ne }\{0\}$, choose $u\in X$ with $\mathrm{\parallel }u\mathrm{\parallel }=1$. Then $\mathrm{\parallel }{\iota }_X(u){\mathrm{\parallel }}_{\max }=1$, and also $\mathrm{\parallel }{\pi }_X(u,0)\mathrm{\parallel }=1=\mathrm{\parallel }(u,0){\mathrm{\parallel }}_{\max }$, so $\mathrm{\parallel }{\iota }_X\mathrm{\parallel }=\mathrm{\parallel }{\pi }_X\mathrm{\parallel }=1$. If $X=\{0\}$, then both ${\iota }_X$ and ${\pi }_X$ are the zero operator, so both norms are $0$. The same argument with $Y$ gives the corresponding statements for ${\iota }_Y$ and ${\pi }_Y$.

Therefore ${\pi }_X\in \mathcal{B}(X\times Y,X)$, ${\pi }_Y\in \mathcal{B}(X\times Y,Y)$, ${\iota }_X\in \mathcal{B}(X,X\times Y)$, and ${\iota }_Y\in \mathcal{B}(Y,X\times Y)$, with the norms stated above.