
Orthogonal Complement 1
Dear Colleagues,
Could you please help me in solving the following problem:
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.

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 $\displaystyle e_i$, $\displaystyle i=1,\ldots, n$.

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.

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.

Thank you very much for your reply. But why this function.

What about $\displaystyle \ker f$ ?