if $W$ is a subspace of an inner product space $V$, which of the following statements are true?
$1)$ there is a unique subspace $W'$ such that $W' + W = V$
$2)$ there is a unique subspace $W'$ such that $W'\oplus W = V$
$3)$ there is a unique subspace $ W'$ such that $W' + W = V$ and $\langle w, w '\rangle = 0 $ for all $w \in W $ and $w' \in W' $
$4) $ there is a unique subspace $W$' such that $W' \oplus W = V $ and $ \langle w,w '\rangle $ = $0$ for all $ w \in W$ and $w' \in W' $
I thinks all options $1,2,3,4 $ will be true because $W' \cap W = \{0\}$
Any Hints/solution
Thanks u
Best Answer
(1) and (2) are certainly false; in fact they're false if $V=\Bbb R^2$ and $W=\{x,0)::x\in\Bbb R\}$.
(3) and (4) are true if $V$ has finite dimension (or if $V$ is a Hilbert space and $W$ is a closed subspace), but they're also false in a general inner-product space.
For example, let $V$ be the space of sequences $x=(x_1,\dots)$ such that all but finitely many of the $x_j$ vanish, with inner product $$(x,y)=\sum x_jy_j.$$Let $W=\{x\in V:\sum x_j=0\}$.
(Users who said (3) and (4) were true presumably had $W'=W^\perp$ in mind. But here it's easy to see that $W^\perp=\{0\}$.)
Similarly if $V$ is a Hilbert space and $W$ is any non-closed subspace: $$W\oplus W^\perp\ne\overline W\oplus W^\perp =\overline W\oplus\overline W^\perp=V.$$