Let $\mathbb{S} = \{\ { P \in \mathbb{R}_2[X] : P(1) = 0\ } \}\subset \mathbb{R}_2[X].$
Notation clarification: $\mathbb{R}_2[X]$ stands for: Span of {$1 ,x, x^2$}, polynomials of degree $\leq$ $2$, of real coefficients.
a) Find all $a \in \mathbb{R}$ for which there exists an inner product in $\mathbb{R}_2[X]$ satisfying:
$\mathbb{S^\perp}$ $=$ $< x^2 + ax -2> $
b) For $a=2$, find an inner product which satisfies the condition mentioned in item "a)", and also:
$||x-1||$ = 1
$||x^2-1||$ = 1
$d(-4x+3, \mathbb{S}) = 3$ ;
Where $d(v, \mathbb{S})$ is the distance from an element $v \in \mathbb{R}_2[X]$ to the subspace $\mathbb{S}$ defined by $d(v, \mathbb{S}) = ||v-T_s(v)|| = ||T_{s^{\perp}}(v)||$, where $T_s$ is the orthogonal proyection operator.
I'm mainly having trouble with part a), I can't seem to find a restriction for $a$. I know an inner product is well-defined over a basis, but I'm not sure I understand under what conditions is not possible for an inner product to exist.
Any hints or help is greatly appreciated. Many thanks in advance.
Hint (for a): such an inner product will exist if and only if $x^2 + ax - 2$ is not an element of $\Bbb S$.