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I was trying to understand why or where would the problem arise in the definition of the direct sum for the coproduct/direct sum for the set (1, 1, 1, .....) infinite number of times. I was trying to reason out as follows. I posted this on the set theory but I am not sure how to tag it to algebra hence I decided to copy paste the same.

(1, 1, 1, ......)[itex]\rightarrow[/itex] Y

[itex]\uparrow[/itex]

f[itex]_{i}[/itex][itex]\rightarrow[/itex] Y

Pls note that the Y is the same as I cannot write the angular arrow.

Now if I say that f[itex]_{i}[/itex] acting on (e[itex]_{i}[/itex]) maps it to ( 0, 0, ...1 at the ith coordinate , 0, ....) then what is the place where I am making a mistake. The problem as I see is that either the map from the set (1, 1, 1, ...) to Y is either not unique or map from f[itex]_{i}[/itex][itex]\rightarrow[/itex] Y does not give the same value as the other path. I am really not sure which is the one and why.

Now Y can be any kind of general space. So I first thought if Y was just the {0} or the whole space R[itex]^{\infty}[/itex]. But I see very clearly that {0} choice does not give a counter example. Not sure how to handle the second case. Even thought with Y= N ( countable ) but still could not get to any kind of solution.

Thx

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# Coproduct , direct sum

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