Regarding $\ell_p$ direct sums

Question

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 Banach 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?

Answer 1

Yes, $X_\alpha$ can be nonzero for uncountably many $\alpha$. Of course, each individual vector $x \in X$ is zero except for countably many coodinates, in order for $\sum_{\alpha\in\Lambda}\|x_\alpha\|^p<\infty$.

added
Suppose (for purposes of contradiction) $x \in X$ is such that there are uncontably many $\alpha$ with $\|x_\alpha\| \ne 0$. Then for some $\varepsilon > 0$ there are infinitely many $\alpha$ with $\|x_\alpha\| > \varepsilon$. [Reason: in fact, $\varepsilon$ may be chosen rational, and use the fact that a countable union of finite sets is countable.]
Let $\alpha_1, \alpha_2,\dots$ be such that $\|x_{\alpha_k}\| > \varepsilon$. Then $$ \sum_{\alpha\in \Lambda}\|x_\alpha\|^p \ge \sum_{k=1}^\infty \|x_{\alpha_k}\|^p > \sum_{k=1}^\infty \varepsilon = +\infty . $$ Thus $x \notin X$.