Real analysis qualifying exam

August 2024
Each problem is worth ten points. Work each problem on a separate piece of paper. If you are not sure whether or not you are allowed to use a particular result to solve a problem, ask. d x d x dxd xdx denotes the Lebesgue measure.
  1. Let f ∈ L + ( X , M , μ ) f ∈ L + ( X , M , μ ) f inL^(+)(X,M,mu)f \in L^{+}(X, \mathcal{M}, \mu)f∈L+(X,M,μ) be a non-negative measurable function on a measure space.
    (a) Show that if ∫ f d μ < ∞ ∫ f d μ < ∞ int fd mu < oo\int f d \mu<\infty∫fdμ<∞, then f < ∞ f < ∞ f < oof<\inftyf<∞ a.e.
    (b) Show that if ∫ f d μ = 0 ∫ f d μ = 0 int fd mu=0\int f d \mu=0∫fdμ=0, then f = 0 f = 0 f=0f=0f=0 a.e.
  2. Let f ( x , t ) : [ 0 , 1 ] × [ 0 , 1 ] → R f ( x , t ) : [ 0 , 1 ] × [ 0 , 1 ] → R f(x,t):[0,1]xx[0,1]rarrRf(x, t):[0,1] \times[0,1] \rightarrow \mathbb{R}f(x,t):[0,1]×[0,1]→R be a measurable function such that for every x ∈ [ 0 , 1 ] x ∈ [ 0 , 1 ] x in[0,1]x \in[0,1]x∈[0,1], the mapping t ↦ f ( x , t ) t ↦ f ( x , t ) t|->f(x,t)t \mapsto f(x, t)t↦f(x,t) is continuous, and furthermore there exists a function g ∈ L 1 ( [ 0 , 1 ] , d x ) g ∈ L 1 ( [ 0 , 1 ] , d x ) g inL^(1)([0,1],dx)g \in L^{1}([0,1], d x)g∈L1([0,1],dx) such that for each ( x , t ) ∈ [ 0 , 1 ] × [ 0 , 1 ] , | f ( x , t ) | ≤ g ( x ) ( x , t ) ∈ [ 0 , 1 ] × [ 0 , 1 ] , | f ( x , t ) | ≤ g ( x ) (x,t)in[0,1]xx[0,1],|f(x,t)| <= g(x)(x, t) \in[0,1] \times[0,1],|f(x, t)| \leq g(x)(x,t)∈[0,1]×[0,1],|f(x,t)|≤g(x). Show that
h ( t ) = ∫ 0 1 f ( x , t ) d x h ( t ) = ∫ 0 1   f ( x , t ) d x h(t)=int_(0)^(1)f(x,t)dxh(t)=\int_{0}^{1} f(x, t) d xh(t)=∫01f(x,t)dx
is continuous.
3. Let X X XXX and Y Y YYY be topological spaces, and X × Y X × Y X xx YX \times YX×Y their product space with the product topology. Denote B X , B Y , B X × Y B X , B Y , B X × Y B_(X),B_(Y),B_(X xx Y)\mathcal{B}_{X}, \mathcal{B}_{Y}, \mathcal{B}_{X \times Y}BX,BY,BX×Y the corresponding Borel σ σ sigma\sigmaσ-algebras. Show that if A ∈ B X A ∈ B X A inB_(X)A \in \mathcal{B}_{X}A∈BX and B ∈ B Y B ∈ B Y B inB_(Y)B \in \mathcal{B}_{Y}B∈BY, then A × B ∈ B X × Y A × B ∈ B X × Y A xx B inB_(X xx Y)A \times B \in \mathcal{B}_{X \times Y}A×B∈BX×Y.
4. Let F F FFF be a Lipschitz continuous function on R R R\mathbb{R}R, that is, | F ( x ) − F ( y ) x − y | ≤ M F ( x ) − F ( y ) x − y ≤ M |(F(x)-F(y))/(x-y)| <= M\left|\frac{F(x)-F(y)}{x-y}\right| \leq M|F(x)−F(y)x−y|≤M for all x ≠ y x ≠ y x!=yx \neq yx≠y. Recall that this implies that F ′ F ′ F^(')F^{\prime}F′ exists a.e. on R R R\mathbb{R}R. Show that for any a < b a < b a < ba<ba<b,
∫ a b F ′ ( x ) d x = F ( b ) − F ( a ) ∫ a b   F ′ ( x ) d x = F ( b ) − F ( a ) int_(a)^(b)F^(')(x)dx=F(b)-F(a)\int_{a}^{b} F^{\prime}(x) d x=F(b)-F(a)∫abF′(x)dx=F(b)−F(a)
You are not allowed to refer to the fact that this conclusion holds for any absolutely continuous function.
5. Let
C 0 , 0 [ 0 , 1 ] = { f ∈ C ( [ 0 , 1 ] , R ) : f ( 0 ) = f ( 1 ) } C 0 , 0 [ 0 , 1 ] = { f ∈ C ( [ 0 , 1 ] , R ) : f ( 0 ) = f ( 1 ) } C_(0,0)[0,1]={f in C([0,1],R):f(0)=f(1)}C_{0,0}[0,1]=\{f \in C([0,1], \mathbb{R}): f(0)=f(1)\}C0,0[0,1]={f∈C([0,1],R):f(0)=f(1)}
Let P 0 , 0 P 0 , 0 P_(0,0)P_{0,0}P0,0 be the subset of polynomials in C 0 , 0 [ 0 , 1 ] C 0 , 0 [ 0 , 1 ] C_(0,0)[0,1]C_{0,0}[0,1]C0,0[0,1]. Prove that P 0 , 0 P 0 , 0 P_(0,0)P_{0,0}P0,0 is dense in C 0 , 0 [ 0 , 1 ] C 0 , 0 [ 0 , 1 ] C_(0,0)[0,1]C_{0,0}[0,1]C0,0[0,1] in the uniform topology.
6. Show that a normed space X X XXX is complete if and only if any absolutely convergent series is convergent (that is, whenever ∑ ‖ x n ‖ < ∞ ∑ x n < ∞ sum||x_(n)|| < oo\sum\left\|x_{n}\right\|<\infty∑‖xn‖<∞, the series ∑ x n ∑ x n sumx_(n)\sum x_{n}∑xn converges in X X XXX ).
7. Let X X XXX and Y Y YYY be Banach spaces. If T : X → Y T : X → Y T:X rarr YT: X \rightarrow YT:X→Y is a linear map such that f ∘ T ∈ X ∗ f ∘ T ∈ X ∗ f@T inX^(**)f \circ T \in X^{*}f∘T∈X∗ for every f ∈ Y ∗ f ∈ Y ∗ f inY^(**)f \in Y^{*}f∈Y∗, show that T T TTT is bounded.
8. Let X X XXX be a Banach space, V ⊂ X V ⊂ X V sub XV \subset XV⊂X a closed subspace, and x ∈ X ∖ V x ∈ X ∖ V x in X\\Vx \in X \backslash Vx∈X∖V.
(a) Prove that there exists a linear functional ϕ x , V ∈ X ∗ ϕ x , V ∈ X ∗ phi_(x,V)inX^(**)\phi_{x, V} \in X^{*}ϕx,V∈X∗ such that ϕ x , V | V = 0 , ‖ ϕ x , V ‖ = 1 ϕ x , V V = 0 , ϕ x , V = 1 phi_(x,V)|_(V)=0,||phi_(x,V)||=1\left.\phi_{x, V}\right|_{V}=0,\left\|\phi_{x, V}\right\|=1ϕx,V|V=0,‖ϕx,V‖=1, and ϕ x , V ( x ) = inf y ∈ V ‖ x − y ‖ ϕ x , V ( x ) = inf y ∈ V   ‖ x − y ‖ phi_(x,V)(x)=i n f_(y in V)||x-y||\phi_{x, V}(x)=\inf _{y \in V}\|x-y\|ϕx,V(x)=infy∈V‖x−y‖
(b) Suppose X X XXX is a Hilbert space, and V V VVV has an orthonormal basis { v i : i ∈ I } v i : i ∈ I {v_(i):i in I}\left\{v_{i}: i \in I\right\}{vi:i∈I}. Find a formula for ϕ x , V ϕ x , V phi_(x,V)\phi_{x, V}ϕx,V.
9. Let ( X , M , μ ) ( X , M , μ ) (X,M,mu)(X, \mathcal{M}, \mu)(X,M,μ) be a finite measure space, and 1 < p < ∞ 1 < p < ∞ 1 < p < oo1<p<\infty1<p<∞. Let f , f n ∈ L p ( X , d μ ) f , f n ∈ L p ( X , d μ ) f,f_(n)inL^(p)(X,d mu)f, f_{n} \in L^{p}(X, d \mu)f,fn∈Lp(X,dμ) for n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N be functions such that f n → f f n → f f_(n)rarr ff_{n} \rightarrow ffn→f pointwise a.e. and sup n ‖ f n ‖ p < ∞ sup n   f n p < ∞ s u p_(n)||f_(n)||_(p) < oo\sup _{n}\left\|f_{n}\right\|_{p}<\inftysupn‖fn‖p<∞. Show that f n → f f n → f f_(n)rarr ff_{n} \rightarrow ffn→f weakly. You may use without proof that an integrable function is uniformly integrable. Comment: the result holds in general measure spaces, but you are not asked to prove that.
10. Give examples of the following. Justify your answers.
(a) A Banach space X X XXX, a closed subspace V V VVV, and a point x ∈ X x ∈ X x in Xx \in Xx∈X such that
‖ x − y ‖ = inf z ∈ V ‖ x − z ‖ ‖ x − y ‖ = inf z ∈ V   ‖ x − z ‖ ||x-y||=i n f_(z in V)||x-z||\|x-y\|=\inf _{z \in V}\|x-z\|‖x−y‖=infz∈V‖x−z‖
for multiple y ∈ V y ∈ V y in Vy \in Vy∈V.
(b) A bounded linear bijection between normed spaces which is not a homeomorphism.
(c) A bounded linear functional on ℓ ∞ ℓ ∞ ℓ^(oo)\ell^{\infty}ℓ∞ which does not arise from duality with ℓ 1 ℓ 1 ℓ^(1)\ell^{1}ℓ1.