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Let be $V$ a vector space, with arbitrary dimension, and $Z\subset V$ proper subspace. Is it always possible write $V$ as a direct sum,

$$V=Z+W?$$

Proof: Let be $\{z_\alpha\}_{\alpha\in \lambda}$ a Hamel basis of $Z$, then since $Z$ is proper subspace of $V$, we can complete $\{z_\alpha\}_{\alpha\in \Lambda}$ to a Hamel basis of $V$, $B=\{\{z_\alpha\}_{\alpha\in \Lambda},\{w_\beta\}_{\beta\in\Gamma}\}$, the construction is similar to the proof of Hamel basis existence.

Take $$W:=span[\{w_\beta\}_{\beta\in\Gamma}]$$

the space generated by finite linear combinations of $w_\beta,$ and the sum is direct by construction.

Is it right?

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    $\begingroup$ Not quite. You wrote "Let … a Hamel basis of $V$", you meant "of $Z$". Apart from that typo, it's correct. $\endgroup$ – Daniel Fischer Jan 24 '16 at 16:55
  • $\begingroup$ I will fix that, thank you!! $\endgroup$ – Irddo Jan 24 '16 at 16:55

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