Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_2$: $$\|f\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$$ 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 $\exists M \geq 0, \forall f \in E, \|T(f)\|_2 \leq M\|f\|_2$. For this, you may consider the family $(f_n)_{n \geq 2}$ of elements of $E$ such that: (i) $f_n$ is piecewise affine, (ii) $f_n(0) = f_n\left(\frac{1}{2} - \frac{1}{n}\right) = f_n\left(\frac{1}{2} + \frac{1}{n^2}\right) = f_n(1) = 0$ and $f_n\left(\frac{1}{2}\right) = 1$.
Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_2$:
$$\|f\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$$
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 $\exists M \geq 0, \forall f \in E, \|T(f)\|_2 \leq M\|f\|_2$. For this, you may consider the family $(f_n)_{n \geq 2}$ of elements of $E$ such that:
(i) $f_n$ is piecewise affine,
(ii) $f_n(0) = f_n\left(\frac{1}{2} - \frac{1}{n}\right) = f_n\left(\frac{1}{2} + \frac{1}{n^2}\right) = f_n(1) = 0$ and $f_n\left(\frac{1}{2}\right) = 1$.