Orthogonal Complement 1

• April 1st 2011, 01:09 PM
raed
Orthogonal Complement 1
Dear Colleagues,

Show that $Y=\{x| \ x=(x_{j})\in \ell^{2}, x_{2n}=0, n=1,2,3,...\}$ is a closed subspace of $\ell^{2}$ and find $Y^{\bot}$.
What is $Y^{\bot}$ if $Y=span\{e_{1},...,e_{n}\}\subset \ell^{2}$, where $e_{j}=(\delta _{jk})$?

Regards,

Raed.
• April 1st 2011, 01:19 PM
girdav
We define a inner product in $l^2$ by $\displaystyle \langle (x_j),(y_j)\rangle := \sum_{j=0}^{+\infty}\overline{x_j}y_j$. By linearity, to be orthogonal to $Y$ is equivalent to by orthogonal to each $e_i$, $i=1,\ldots, n$.
• April 1st 2011, 01:25 PM
raed
Thank you very much for your reply but how can I prove that Y with zero even terms is closed and what its orthogonal complement.
• April 1st 2011, 01:31 PM
girdav
Let $f : (x_0,\ldots,x_n,\ldots)\mapsto (x_0,x_2,\ldots,x_{2n},\ldots )$. Show that $f$ is linear and continuous.
• April 1st 2011, 02:00 PM
raed
What about $\ker f$ ?