**Absolute norm**

Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that $$ \|(x,y)\|_N=N((\|x\|, \|y\|)) \qquad \text{ for all } z=(x,y)\in Z. $$

For example, the $\ell_p$-norms are absolute norms.

**1-unconditional sum**

Let $E$ be a Banach space with a 1-unconditional normalized Schauder basis. We can think of the elements of $E$ as sequences with the property that $$ \|(a_1,a_2,\dots)\|_E=\|(|a_1|,|a_2|,...\|_E \qquad \text{ for all } (a_j)\in E. $$ Note that $E$ is naturally endowed with the structure of a Banach lattice with respect to the pointwise operations.

Suppose that $X_1, X_2,\dots$ are Banach spaces. Their $E$-sum $X=(X_1, X_2, \dots)_E$ consists of all sequences $(x_j)$ with $x_j\in X_j$ and $(\|{x_j}\|)\in E$ with the norm $\|(x_j)\|=\|(\|x_j\|)\|_E$.

**Question**

Let $Z=X_1\times X_2\times...$. Can I equip $Z$ with an absolute norm? If so is this norm equivalent to equipping $Z$ with an 1-unconditional norm?

Thanks in advance!