Let $T>0$ and $f\in C(\mathbb R, L^{2}(\mathbb R))$ with the following property:
Put $g(t):= \sup\limits_{0\leq \tau\leq t} \|f(\tau)\|_{X},$ where $X \subset L^{2}$ and $X$ is a Banach Space.
For $0\leq t \leq T, $ $0<p<\infty,$ $g(t)^p \leq 1+ \int_{0}^{t} g(\tau)^{p} d\tau.$
My Question is: Can we expect to show that $g\in L^{\infty}([0, T])$?
My Vague Idea: I think, we need to use Gronwall Lemma for the integral form, but I am little confuse which form I should consider, and how to apply.