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{p in P2: p(0) > p(1)} {p in P2: p(3) = p(4)} {p in P2: p'(3) = 4p(7)}

I understand that generally we need to check to make sure that these are closed under addition and scalar multiplication, and that it is not an empty set. I am having a very difficult time understanding exactly how to show this though, especially because of the inequality in the first question and the derivative in the third question.

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The zero polynomial isn't in the first subset so it isn't a subspace of $P_2$. The third subset isn't empty since it contains the zero polynomial. Now let $p$ and $q$ in this subset and $\lambda\in\Bbb R$ then

$$(p+\lambda q)'(3)=p'(3)+\lambda q'(3)=4p(7)+4\lambda q(7)=4(p+\lambda q)(7)$$ hence $p+\lambda q$ is in the third subset. Conclude.

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