grandes-ecoles 2020 Q23

grandes-ecoles · France · centrale-maths2__mp Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$. Show that we define an inner product on $E_1$ by setting $$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t)\,\mathrm{d}t$$

Let $E_1$ denote the vector space of functions $f:[0,1] \rightarrow \mathbb{R}$ continuous, of class $\mathcal{C}^1$ piecewise, and satisfying $f(0) = f(1) = 0$.
Show that we define an inner product on $E_1$ by setting
$$\forall (f,g) \in (E_1)^2, \quad (f \mid g) = \int_0^1 f'(t) g'(t)\,\mathrm{d}t$$

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