Let $C_c(\mathbb{R})$ be the following:
$$C_c = \{ f \in C(\mathbb{R}) \mid \exists T > 0 \text{ s.t. } f(t) = 0 \text{ for } |t| \geq T\}$$
Let $T_n \in L(C_c(\mathbb{R}))$ be a linear operator such that:
$$T_n u = \delta_n *u, \forall u \in C_c(\mathbb{R}),$$
where
$$ \delta_n(t)= \begin{cases} n^2(t+1/n) & -1/n \leq t \leq 0 \\ -n^2(t-1/n) & 0 < t \leq 1/n \\ 0 & \text{elsewhere} \end{cases} $$
I have to prove that with respect to the 2-norm, the operator $T_n$ has $\|T_n\| = 1.$
I cannot use the Young's convolution inequality directly, but I have to use the properties of the Fourier's transform.
So far, I was trying to do the following:
$$\|\delta_n * u\|_2 = \|\mathfrak{F}(\delta_n * u)\|_2 = \|\mathfrak{F}(\delta_n ) \mathfrak{F}(u)\|_2 $$
But then I was wondering how to proceed.