Let $ {P_{2}}([0,1]) $ be the Hilbert space consisting of all polynomials of degree at most $ 2 $ (including the zero polynomial on $ [0,1] $) equipped with the inner product $ \displaystyle \langle f,g \rangle \stackrel{\text{def}}{=} \int_{0}^{1} f(x) g(x) ~ d{x} $. Define a linear functional on $ {P_{2}}([0,1]) $ by $ \phi(p) \stackrel{\text{def}}{=} p(1) $ for all $ p \in {P_{2}}([0,1]) $.
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How can I show that $ \phi $ is linear, and how can I find $ \| \phi \|$? I guess I
don’t quite understand what $ \phi(p) = p(1) $ means. As for the norm $ \| \cdot \| $
, can I use the assertion of the Riesz Representation Theorem that the operator norm
of the linear functional is equal to the vector norm of the vector that induces it? -
Find the polynomial $ q \in {P_{2}}([0,1]) $ such that $ \phi(p) = \langle p,q
\rangle $ for all $ p \in {P_{2}}([0,1]) $. I know that for this part, a unique such
polynomial $ q $ exists by the Riesz Representation Theorem and that I have to get an
orthonormal basis for $ {P_{2}}([0,1]) $. But I’m stuck from there. Appreciate any
help, dear colleagues.
Best Answer
Just a hint:
$\phi(p +aq) = p(1) + aq(1) = \phi(p) + a\phi(q)$ therefore $\phi$ is linear.
$||\phi|| = \sup_{p : \int_0^1p^2 = 1}\phi(p) = \sup_{p : \int_0^1p^2 = 1}p(1)$.
$p = ax^2 + bx +c$, $\int_0^1p^2 = 1 \Longleftrightarrow \frac{a^2}{5} + \frac{ab}{2} + \frac{2ac+b^2}{3} + bc + c^2 = 1$.
Now you have to maximize $p(1) = a +b + c$ bearing in mind the previous condition.
Also: $p = ax^2 + bx +c$, $q = dx^2 + ex +f$. $\int_0^1pq = a(\frac{d}{5} + \frac{e}{4} + \frac{f}{3}) + b(\frac{d}{4} + \frac{e}{3} + \frac{f}{2}) + c(\frac{d}{3} + \frac{e}{2} + f)$. You are looking for $q$ such that the previous integral equals $p(1) = a + b +c$, so you look for $d,e,f$ such that $\frac{d}{5} + \frac{e}{4} + \frac{f}{3} = 1$, $\frac{d}{4} + \frac{e}{3} + \frac{f}{2} = 1$ and $\frac{d}{3} + \frac{e}{2} + f = 1$.