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Let $X$ and $Y$ be two Banach spaces with norms $\|\|_X$and

Let $X$ and $Y$ be two Banach spaces with norms $\|\|_X$ and $\|\|_Y$ respectively. If $Z=X\times Y$ is also a Banach space with norm $\|\|_Z$ then what is the relation between $\|\|_X,\:\|\|_Y$ and $\|\|_Z$?

Let $X$ and $Y$ be two Banach spaces, and $\|\cdot\|\_X$, $\|\cdot\|\_Y$ and $\|\cdot\|\_{X\times Y}$ are norms on $X,Y$ and $X\times Y$ respectively. Then my problem is that what is the relation between $\|\cdot\|_{X\times Y}$ and $\|\cdot\|\_X,\|\cdot\|_Y$?

Let $X$ and $Y$ be two Banach spaces with norms $\|\|_X$and

Let $X$ and $Y$ be two Banach spaces and $\|\|_X,\|\|_Y$ and $\|\|_{X\timesY}$ are norms on $X,Y$ and $X\timesY$ respectively. Then my problem is that what is the relation between $\|\|_{X\timesY}$ and $\|\|_X,\|\|_Y$?

Let $X$ and $Y$ be two Banach spaces, and $\|\cdot\|\_X$, $\|\cdot\|\_Y$ and $\|\cdot\|\_{X\times Y}$ are norms on $X,Y$ and $X\times Y$ respectively. Then my problem is that what is the relation between $\|\cdot\|_{X\times Y}$ and $\|\cdot\|\_X,\|\cdot\|_Y$?