Definition of direct sum of Banach spaces

Question

Given a family $\{A_i\}_{i\in I}$ of Banach spaces, can anyone please provide me with the definitions of the following concepts

i) $c_0-$ direct sum

ii) $l^{\infty}$ direct sum

iii) algebraic direct sum

What's difference between these three concepts? Can anyone please elaborate with an example taking a particular family of Banach spaces $\{B(H_i)\}_{i \in I}$ where $H_i$'s are Hilbert spaces of finite dimension, say, $n_i$?

Answer 1

It appears be that (i) requires a sequence of spaces $(E_n)_{n\in\Bbb N}$:

(i) $c_0$ direct sum: $$\{(x_n)_{n\in\Bbb N}\in\prod_{n\in\Bbb N}E_n :\lim_{n\to\infty}\|x_n\| = 0\}.$$

(ii) $l^\infty$ direct sum: $$\{(x_i)_{i\in I}\in\prod_{i\in I}E_i :\sup_{i\in I}\|x_i\|<\infty\}.$$

(iii) Algebraic direct sum: $$\{(x_i)_{i\in I}\in\prod_{i\in I}E_i : x_i\ne 0\hbox{ for only a finite number of indices $i$}\}.$$

Source for (i) and (ii) : Measure Theory and Functional Analysis

Answer 2

For general (non-sequnce) case of (i) we want the set of all
$(x_i)_{i\in\Bbb I}\in\prod_{i\in I}E_i$ such that for all $\epsilon >0$, the set $\{i \in I : \|x_i\| > \epsilon\}$ is finite.