# Convergence in Lebesgue-Bochner Space $L^{\infty}(0,T,L^1(\Gamma))$

Let $$\Gamma$$ be a compact $$C^2$$ manifold and suppose that $$f_n$$ is a non negative sequence of functions such that $${\vert \vert f_n \vert \vert}_{L^{\infty}(0,T,L^1(\Gamma))} \le C$$

I am interested in deducing convergence of $$f_n$$

MY ATTEMPTS:

1. Since $$L^1$$ is not reflexive, $$L^{\infty}(0,T,L^1(\Gamma))$$ is also not reflexive and thus from the boundedness of $$f_n$$ I can't obtain a weak convergent subsequence.
2. After that, I wondered if I could have a weak-* convergence so I thought the Banach-Alaoglu theorem. But again, this didn't work because I couldn't find the Banach and separable space whose dual is $$L^{\infty}(0,T,L^1(\Gamma))$$

2nd EDIT: I just came up with the following idea for which I need also verification:

Consider the space of continuous functions with compact support on $$\Gamma$$, i.e $$C_c(\Gamma)$$. Since $$\Gamma$$ is compact, we know that $$C_c(\Gamma)$$ is also Banach and separable. Its dual space is the space of (signed) Radon measures on $$\Gamma$$ with finite mass which is denoted by $$\mathcal M(\Gamma)$$.

If $$L^{\infty}(0,T,\mathcal M(\Gamma))$$ is contained in the dual space of $$L^1(0,T,\mathcal C_c(\Gamma))$$ then by Banach-Alaoglu theorem a weak-* convergent subsequence is obtained.

However I'm not completely sure if the duality argument that I used holds.

At this point I've been stuck. I would really appreciate any help or even hints.

• If it were $L^\infty(\Omega)$, where $\Omega$ is a $\sigma$-finite measure space, then $L^\infty$-boundedness would give you the existence of a subsequence that converges weakly-$\star$. Surely there is a version of this result for the spaces you are working with. (I see that you added this in your edit. Try looking in these lecture notes of John Hunter: – Giuseppe Negro Jan 23 at 12:24
• @GiuseppeNegro My only doubt is if $\mathcal M(\Gamma)$ is a separable space in order for the duality argument to be valid. I saw in your answer in that post you recommend a book in evolution equations. I'll chech it now. Thanks a lot – kaithkolesidou Jan 23 at 12:32
The dual space of $$L^1(0,T; C(\Gamma))$$ is $$L^\infty_w(0,T;\mathcal M(\Gamma))$$ and this space consists of weak-$$*$$ measurable functions, see, e.g., Theorem 10.1.16 in "Handbook of applied analysis" by Papageorgiou and Kyritsi-Yiallourou.