(Solved): Let \( \mathbb{C}[x] \) denote the set of all polynomials in a complex variable \( x \) and with co ...

Let \( \mathbb{C}[x] \) denote the set of all polynomials in a complex variable \( x \) and with coefficients in \( \mathrm{C} \). (i) Show that \( \mathbb{C}[x] \) is a vector space over \( \mathbb{C} \). (ii) Prove that \[ \langle f, g\rangle=\int_{-1}^{1}\left(14 x^{2} f(x) \cdot \overline{g(x)}+12 f^{\prime}(x) \cdot \overline{g^{\prime}(x)}\right) d x \] is an inner product on \( \mathbb{C}[x] \). (iii) Show that for any \( f \in \mathrm{C}[x] \) we have \[ \left|\int_{-1}^{1}\left(14 x^{4} f(x)+24 x f^{\prime}(x)\right) d x\right|^{2} \leq 36\|f\|^{2}, \] where \( \|f\|=\langle f, f\rangle^{\frac{1}{2}} \),