TITLE: How to 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 $D?$
QUESTION [5 upvotes]: 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 $D$.

REPLY [4 votes]: Take infinite dimensional  $\ell_2$ and: 
$\ \ \ A=\{ (x_i) : x_i=0, i\text{ even}\}$,
$\ \ \ B=\{ (x_i) : x_i=0, i\text{ odd}\}$, 
$\ \ \ C=\{ (x_i) : x_1=0\}$,
$\ \ \ D=\{ (x_i) : x_i=0, i>1\}$.