Let $V$ be vector space of countable dimension over field $K$. Is it true that $V\cong V\oplus V$?

Question

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.

They must mean "countably infinite" when they say countable. Some people do this. — Randall, Jul 09 '18 at 18:46
@DaveWasHere Could you post a link with the answer for the infinite case? Thanks! — 1123581321, Jun 17 '19 at 10:40
This post was essentially a (multi)duplicate: Dimension of Direct sum of same Vector Spaces. — Anne Bauval, Oct 04 '23 at 14:52

score 9 · Accepted Answer · edited Jul 09 '18 at 19:13

9

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

edited Jul 09 '18 at 19:13

Ivo Terek

80,301

answered Jul 09 '18 at 18:50

Paul Frost

87,968

2

Yes, exactly. At least I'm not crazy... So surely for the authors "countable = countably infinite". – DaveWasHere Jul 09 '18 at 18:52
1

Yes, as Randall said. – Paul Frost Jul 09 '18 at 18:54
This post was essentially a (multi)duplicate: Dimension of Direct sum of same Vector Spaces. – Anne Bauval Oct 04 '23 at 14:53

Let $V$ be vector space of countable dimension over field $K$. Is it true that $V\cong V\oplus V$?

1 Answers1