Question about isomorphic mapping on direct sums?

[Ishida52134]

Main Question or Discussion Point

The identity map on the direct sum of V1 and V2 would be i1 composed with p1 + i2 composed with p2. Would such an identity map exist for an infinite direct sum? And an analogous mapping for a direct product?

 

[economicsnerd]

You're given:
- $V = \bigoplus_{a\in A} V_a$, where $A$ is some possibly infinite index set.
- For each $a\in A$, the inclusion map $\iota_a: V_a \to V$ given by $\iota_a(x_a) = (x_a, (0_b)_{b\in A\setminus\{a\}})$.
- For each $a\in A$, the projection map $\pi_a: V \to V_a$ given by $\pi_a(x) = (x_a)$.

You've noticed that, if $|A|<\infty$, then $\sum_{a\in A} \iota_a\circ\pi_a = id_V: V\to V$.

So you're asking whether the condition that $|A|<\infty$ can be dropped to draw the same conclusion? The answer is essentially
1) Yes.
2) One has to be careful about the definition of $\sum_{a\in A}$ when $A$ is infinite.
3) There's very little content here, as point (2) is essentially built into the definition of a direct sum.