The question required me to find the maximal value of the function below at unit ball: $$|\int_0^Te^{-(T-τ)}x(τ)d(τ)|$$ for function x ∈ $L_2$ ([0,T ]).
I know we can it by the Cauchy-Schwartz inequality, but how can we write it in the form of the inequality? Can anyone pls help?
Is it correct if I use Cauchy-Schwartz inequality to prove the above function is continuous, then it can attain its maximum value at unit ball. After that, since it is continuous, we can say that it obtain maximum when T=0?