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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.2012-01-23

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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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    @cap, $K^{\infty}\oplus K^{\infty}$ just means $\{\,(a,b):a,b{\rm\ in\ }K^{\infty}\,\}$.2016-06-12