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I've tried to solve the following question (Exercise 10, page 107 from Roman's book: Advanced Linear Algebra), but I wasn't able to solve it.

Find a vector space V and decompositions $V=A\oplus B = C\oplus D$ with $A$ isomorphic to $C$ but $B$ is not isomorphic to $D$.

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Note that clearly $V$ cannot be finite dimensional, then it might be easier to see how to construct an example. – Tobias Kildetoft Jan 23 '12 at 20:32
up vote 3 down vote accepted

Take infinite dimensional $\ell_2$ and:

$\ \ \ A=\{ (x_i) : x_i=0, i\text{ even}\}$,

$\ \ \ B=\{ (x_i) : x_i=0, i\text{ odd}\}$,

$\ \ \ C=\{ (x_i) : x_1=0\}$,

$\ \ \ D=\{ (x_i) : x_i=0, i>1\}$.

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Hello!Sorry, but $A$ and $C$ are not isomorphic as vector spaces. Am I missing something? Thanks! – spohreis Jan 23 '12 at 21:06
@user23505 Sorry, I had my letters mixed up. Thanks for pointing it out. It is correct now. – David Mitra Jan 23 '12 at 21:16
Perhaps one could just say that for any field $K$, $K^{\infty}=K^{\infty}\oplus K^{\infty}=K^{\infty}\oplus K$? – Gerry Myerson Jan 24 '12 at 0:33
@GerryMyerson I don't think $K^{\infty}\oplus K^{\infty}$ is a direct sum since the intersection of the factors is not ${0}$. – cap Jun 12 at 3:12
@cap, $K^{\infty}\oplus K^{\infty}$ just means $\{\,(a,b):a,b{\rm\ in\ }K^{\infty}\,\}$. – Gerry Myerson Jun 12 at 3:17

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