# Orthogonal Complement 1

• Apr 1st 2011, 12:09 PM
raed
Orthogonal Complement 1
Dear Colleagues,

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

Regards,

Raed.
• Apr 1st 2011, 12:19 PM
girdav
We define a inner product in $\displaystyle l^2$ by $\displaystyle \displaystyle \langle (x_j),(y_j)\rangle := \sum_{j=0}^{+\infty}\overline{x_j}y_j$. By linearity, to be orthogonal to $\displaystyle Y$ is equivalent to by orthogonal to each $\displaystyle e_i$, $\displaystyle i=1,\ldots, n$.
• Apr 1st 2011, 12:25 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.
• Apr 1st 2011, 12:31 PM
girdav
Let $\displaystyle f : (x_0,\ldots,x_n,\ldots)\mapsto (x_0,x_2,\ldots,x_{2n},\ldots )$. Show that $\displaystyle f$ is linear and continuous.
• Apr 1st 2011, 01:00 PM
raed
What about $\displaystyle \ker f$ ?