I am reading this paper by S.H Karin titled Norm attaining operators and pseudospectrum. In page 2 he gives the definition of $l_p$ direct sum of a family of Banch spaces as follows: If $1\leq p< \infty$ and $\{X_\alpha\}_{\alpha\in\Lambda}$ is a faily of Banach spaces, then their $l_p$ direct sum is the space $$X=\bigg\{ x\in\prod\limits_{\alpha\in\Lambda}X_\alpha:\sum\limits_{\alpha\in\Lambda}\|x_\alpha\|^p<\infty\bigg\}$$ endowed with the norm $$\|x\|=\left(\sum\limits_{\alpha\in\Lambda}\|x_\alpha\|^p\right)^{\frac{1}{p}}.$$ I want to ask if $X_\alpha$ can be non zero for all $\alpha\in\Lambda$? Can $\Lambda$ be an uncountable set?