Hermitian Adjoint of V & R Vector Spaces Homework

[Doradus]

Homework Statement

Be V the set $\{f \in \mathbb{R}[X]| deg\,f \leq 2 \}$. This becomes to an euclidic vector space through the
inner product $\langle f,g\rangle:=\sum_{i=-1}^1f(i)g(i)$ .
The same goes for $\mathbb{R}$ with the inner product $\langle r,s\rangle :=rs\,\,\,$.

a) For $j:\mathbb{R}\to V,r\mapsto rX$, calculate the hermitian adjoint $j^*$.

b) Be $\Phi :V \to \mathbb{R}$ the linear map $\sum_{i=0}^2a_iX^i \mapsto \sum_{i=0}^2a_i \,\,\,$. Calculate the hermitian adjoint $\Phi^*\,\,\,$.

Homework Equations

The Attempt at a Solution

For a) i have the follwowing solution:

$\langle f,j(s) \rangle_V = \langle j^*(f), s \rangle_{\mathbb{R}}$
$\Rightarrow \sum_{i=-1}^1f(i) \cdot (j(s))(i)=j^*(f) \cdot s$
$\Rightarrow f(-1)\cdot -s+f(0)\cdot 0s+f(1)\cdot s = j^*(f) \cdot s$
$\Rightarrow j^*(f)=f(1)-f(-1)$

Is this solution correct?
For b), i don't find a starting point.

 

[andrewkirk]

Can't we approach (b) the same way as (a)?
For (b), the defining equation is
$$\langle\Phi(f),s\rangle=\langle f,\Phi^*(s)\rangle$$
What happens if we expand that using the definitions given?

Your working for (a) looks broadly correct. To check that something has not gone wrong, like a missed sign, plug a polynomial $f(x)=a_0+a_1x+a_2x^2$ into it and see if the equality of the two inner products holds.