課程名稱︰實分析二
課程性質︰數學研究所必選修
課程教師︰劉豐哲
開課學院：理學院
開課系所︰數學系、數學研究所、應用數學科學研究所
考試日期（年月日）︰104.6.24
考試時限（分鐘）：100min
試題 :
2
1.Let E = L (Ω,Σ,μ) ,μ(Ω)<∞ , and F a sub σ-algebra of Σ.
Put M = {[g]∈E : g is F-measurable}.
(7%)(i) Show that M is a closed vector subspace of E.
2 2
(8%)(ii) Suppose that f∈L (Ω,Σ,μ). Show that there is g∈L (Ω,Σ,μ)
such that ∫fdμ = ∫gdμ , A∈F.
A A
2
(15%)2.Suppose that f is AC on [-π,π] and f'∈L[-π,π]. Assume further that
f(-π)=f(π). Show that S (f,t)→f(t) uniformly on [-π,π] when n→∞.
n
(20%)3.Let (Ω,Σ,μ) be a σ-finite measure space and 1<p,q<∞ be conjugate
p
exponents. Suppose that {f } is a sequence in L (Ω,Σ,μ) with the
q
property that lim ∫ f g dμ =∫ fg dμ for all g∈L (Ω,Σ,μ).
n→∞ Ω n Ω
n n
4.For f,g in the Schwartz space S in R , define f＊g(x)=∫f(x-y)g(y)dy ,x∈R .
(9%)(i) Show that for any milti-index α we have
^α n/2 |α| α ^ ^ n
∂(f＊g)(ξ)=(2π) i ξ {f(ξ)}{g(ξ)} , ξ∈R .
(6%)(ii) Show that f＊g∈S if f,g are in S .
(15%)5.Let E be a Hilbert Space and A：E→E be linear . Suppose that (Ax,y)=
(x,Ay) for all x,y in E . Show that A is a bounded liner map .
p
(15%)6.Let {f } be a bounded sequence in L (Ω,Σ,μ) , where μ(Ω)<∞ and
n
1 1
p>1 . Assume that f →f a.e. . Show that f∈L (Ω,Σ,μ) and f →f in L.
n n
∞
(15%)7.Put X=l (N) , the space of all bounded sequences x=(x ,x ,...)=(x )
1 2 n
of real numbers . Let T be the left shift on X , i.e. (Tx) =x , n=1,2,..
n n+1
For n∈N , let c x = (x +...+x )/n , x=(x ) , and define p(x)=limsup c x.
n 1 n n n→∞ n
Then p is a sublinear functional on X . Let now Y={x∈X：lim c x exists}
n→∞ n
and define l(x)=lim c x for x∈Y . Show that l can be extended to be a
n→∞ n
linear functional Λ on X such that Λ(Tx)=Λ(x) for all x∈X .