Question about isomorphic mapping on direct sums?

  • #1
139
0

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?
 

Answers and Replies

  • #2
269
24
You're given:
- [itex]V = \bigoplus_{a\in A} V_a[/itex], where [itex]A[/itex] is some possibly infinite index set.
- For each [itex]a\in A[/itex], the inclusion map [itex]\iota_a: V_a \to V[/itex] given by [itex]\iota_a(x_a) = (x_a, (0_b)_{b\in A\setminus\{a\}})[/itex].
- For each [itex]a\in A[/itex], the projection map [itex]\pi_a: V \to V_a[/itex] given by [itex]\pi_a(x) = (x_a)[/itex].

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

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

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