grandes-ecoles 2014 Q1b

grandes-ecoles · France · x-ens-maths__psi Groups Group Homomorphisms and Isomorphisms

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Calculate the minimal possible value for the constant $M$ in the relation $\exists M \geq 0, \forall f \in E, \|T(f)\|_E \leq M\|f\|_E$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$:
$$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$
We denote by $T$ the application defined on $E$ such that:
$$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$
Calculate the minimal possible value for the constant $M$ in the relation $\exists M \geq 0, \forall f \in E, \|T(f)\|_E \leq M\|f\|_E$.

Paper Questions

Q1a Q1b Q1c Q1d Q1e Q2a Q2b Q2c Q2d Q2e Q3a Q3b Q3c Q3d Q3e Q4a Q4b Q4c Q4d Q4e Q4f Q4g Q4h Q4i