Question about isomorphic mapping on direct sums?

[Ishida52134]

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.