Problem: (1) Narrate the resonance theorem. (2) Let $X$ be a Banach space, and denote by $C_w([0,T];X)$ be all the maps $$ex a{cccc} u:&[0,T]& o& X\ &t&mapsto &u(t) ea eex$$ such that for any functional $phiin X'$, the function $[0,T]
i tmapsto sef{phi,u(t)}$ is continuous. Utilize (1) to show $C_w([0,T];X)subset L^infty([0,T];X)$.
Proof: (1) The resonance theorem reads as follows. Let $X$ be a Banach space, $Y$ be a normed linear space. Suppose that $sed{A_lm}_{lmin vLm}$ is a family of bounded linear map from $X$ to $Y$, and satisfies $$ex sup_{lmin vLm} sen{A_lm x}<infty,quadforall xin X. eex$$ Then there exists a positive constant $M$ such that for any $lmin vLm$, $sen{A_lm}leq M$.
(2) As is well-known, $X$ can be viewed as a bounded linear functional on $X'$; that is, $Xsubset X''$. Let $uin C_w([0,T];X)$. We will use (1) to show $uin L^infty(0,T;X)$. Indeed, for $forall phiin X'$, $sef{phi,u(t)}$ is a continuous function on $[0,T]$, and thus is bounded, i.e., $$ex sup_{tin [0,T]}|sef{phi,u(t)}|<infty,quadforall phiin X'. eex$$ Consequently, $sed{u(t)}_{tin [0,T]}$ viewed as a family of bounded linear functional on $X'$, satisfies the hypothesis of (1), and hence there exists an $M>0$ such that $$ex sup_{tin [0,T]}sen{u(t)}_{X} =sup_{tin [0,T]}sen{u(t)}_{X''} leq M. eex$$