2

Find the orthogonal complement of a subspace $$ M = \{ x \in L_2(-1, 1):x(t)=-x(-t), \int_0^1 x(t)t^2dt=0 \} $$ in $L_2(-1, 1).$

As I understand M can be described as all odd functions which are orthogonal to $ \lambda t^2 $ on $(0, 1)$. But I don`t know how to find the orthogonal complement

s909
  • 43

1 Answers1

2

Let $f(t)=t^{2}\chi_{(0,1)}(t)$. Then $M$ consists of functions which are odd and orthognal to $f$. This means $M$ is precisely the orthogonal complement of the span of even functions and $f$. Thus, $M^{\perp} =\{g+cf:c \in \mathbb R, g \, \text {is even} \}$

Kavi Rama Murthy
  • 311,013
  • 1
    Sorry again. Can you explain please, why the set ${g+cf:c \in \mathbb R, g , \text {is even} }$ is closed? – thing Dec 20 '20 at 13:09
  • @thing That is quite elementary. If you take any closed subspace $M$ and take $x_0 \notin M$ then $span (M \cup {x_0})$ is always a closed subspace. I can give a proof butI I suggest you try to prove it yourself. – Kavi Rama Murthy Dec 20 '20 at 23:16
  • But here we're not talking about the union of closed sets, but about their direct sum. It seems to me that in the general case the direct sum of closed sets will not be closed. If you know about this fact, please give me a link. – thing Dec 21 '20 at 00:19
  • @thing Hint: If $m_n+c_nx_0 \to y$ consider the cases $|c_n| \to \infty$ and $(c_n)$ bounded along a subsequence. In the first case divide by $c_n$ to get a contardiction. In the second case take a convergent subsequence. This is a stanadard argument. – Kavi Rama Murthy Dec 21 '20 at 00:30
  • It's clear now. Thanks. – thing Dec 21 '20 at 01:37