Reflexive Banach Space and Duality 

1. If $X$ is reflexive, then $X^{\ast\ast} = X$ by definition, so we have \[ (X^\ast)^{\ast\ast} = (X^{\ast\ast})^\ast = X^\ast \] so $X^\ast$ is reflexive.

2. Since $X^\ast$ is reflexive, by 1. we have that $X^{\ast\ast}$ is reflexive. But $J_X$ is an isomorphism between $X$ and a closed subspace of $X^{\ast\ast}$, and a closed subspace of a reflexive space is reflexive, so $X$ is reflexive.

3. If there exists $u \in L^\infty(E)$ such that $\int_E uv$ does not take a maximum on the closed unit ball in $L^1(E)$, then the unit ball is not compact in the weak topology and so $L^1(E)$ is not reflexive. (I assume this is the result "seen in class", but I'm guessing here).