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Let $V$ be the vector space of polynomials in the variable $t$ of degree at most 2 over $\mathbb{R}$. An inner product on $V$ is defined by

$$\langle f, g\rangle=\int_{0}^{1} f(t) g(t) d t$$

for $f, g \in V$. Let $W=\operatorname{span}\left\{1-t^{2}, 1+t^{2}\right\}$ and $\mathrm{W}^{\perp}$ be the orthogonal complement of $W$ in $V$. Which of the following conditions is satisfied for all $h \in W^{\perp}$?

1. $h$ is an even function, i.e. $h(t)=h(-t)$

2. $h$ is an odd function, i.e. $h(t)=-h(-t)$

3. $h(t)=0$ has a real solution

4. $h(0)=0$
in Functional Analysis by Expert | 46 views

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