I know that my question has been answered on StackExchange for the countably infinite case. I understand why this holds. However, I'm reading "A course in homological algebra" by Hilton,Stammbach and they assert in an exercise that $V\cong V\oplus V$ holds for countable dimensional (so possibly finite) vector spaces. I can't see why or even if this claim is true.

Please help if you have any idea if this is true or not.

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    $\begingroup$ They must mean "countably infinite" when they say countable. Some people do this. $\endgroup$ – Randall Jul 9 '18 at 18:46
  • $\begingroup$ I don't understand your question that V is direct sum of copies of V which is not true since intersection is not trivial between V and V $\endgroup$ – maths student Jul 9 '18 at 18:46
  • $\begingroup$ @DaveWasHere Could you post a link with the answer for the infinite case? Thanks! $\endgroup$ – 1123581321 Jun 17 '19 at 10:40

For a finite-dimensional $V$ it is obviously wrong because $\dim(V \oplus V) = 2 \dim V > \dim V$.

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    $\begingroup$ Yes, exactly. At least I'm not crazy... So surely for the authors "countable = countably infinite". $\endgroup$ – DaveWasHere Jul 9 '18 at 18:52
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    $\begingroup$ Yes, as Randall said. $\endgroup$ – Paul Frost Jul 9 '18 at 18:54

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