Orthogonal Complement 1

**raed** · April 1st 2011, 12:09 PM

Dear Colleagues,

Could you please help me in solving the following problem:
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.

**girdav** · April 1st 2011, 12:19 PM

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

**raed** · April 1st 2011, 12:25 PM

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.

**girdav** · April 1st 2011, 12:31 PM

Let $f : (x_0,\ldots,x_n,\ldots)\mapsto (x_0,x_2,\ldots,x_{2n},\ldots )$ . Show that $f$ is linear and continuous.

**raed** · April 1st 2011, 01:00 PM

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

**girdav** · April 1st 2011, 01:06 PM

What about $\ker f$ ?

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