Lp Spaces and Sequence Spaces

Prove the following variant of Egorov’s theorem that applies to sets of infinite measure. Let \((X,\mathcal {M},\mu )\) be a measure space and let \(g\in L^1(\mu )\). Suppose \(f_n:X\to \mathbb {R}\) are measurable with \(|f_n|\le g\) and \(f_n\to f\) pointwise. Then for every \(\epsilon >0\) there is a set \(E\subset X\) of measure \(<\epsilon \) so that \(f_n\to f\) uniformly on \(X\setminus E\).

Let \(V,W\) be Banach spaces, and let \(T_i:V\to W\) be a sequence of continuous linear operators such that \(\lim _{i\to \infty }T_i v\) exists for every \(v\in V\). Define the linear map \(T:V\to W\) by \[ Tv=\lim _{i\to \infty }T_i v \]

Show that \(T\) is continuous.

Suppose \(f_i:\mathbb {R}\to \mathbb {R}\) is measurable for \(i=1,2,\cdots \). Is the set \[ \{x\in \mathbb {R}\mid 2\text { is a limit point of }\{f_i(x)\}_{i=1}^{\infty }\} \]

measurable? A number \(t\) is a limit point of a sequence if some subsequence converges to it.

Let \((X,\mathcal {M},\mu )\) be a measure space with \(\mu (X)<\infty \). Show that if \(1\le q< p\le \infty \) then \(L^q(X)\supseteq L^p(X)\). Moreover, if \(X\) contains an infinite family of pairwise disjoint subsets of positive measure, then \(L^q(X)\supsetneq L^p(X)\).

Prove that the dual space of \((c_0,\|\cdot \|_{\sup })\) is \(\ell ^1\). Recall \[ c_0=\{v\in \mathbb {R}^{\mathbb {N}}:\lim _{j\to \infty }v_j=0\} \]

and \(\ell ^1=\{w\in \mathbb {R}^{\mathbb {N}}:\sum _j |w_j|<\infty \}\) with \(\|w\|=\sum _j |w_j|\).

Prove that there exists \(\phi :\ell ^2(\mathbb {N})\to \mathbb {R}\) that is linear and not continuous.

Prove that if \(A:\ell ^2(\mathbb {N})\to \ell ^2(\mathbb {N})\) is continuous and linear and \(AB(0,1)\) is dense in \(B(0,1)\) then \(A\) is surjective. Note that this is true for Banach spaces as well (though we are not asking you to prove it).

We say \(f_1,\ldots \) is \(\lambda \)-Cauchy if for every \(\epsilon >0\) there exists \(N\) so that for \(i,j>N\) we have

Show that if \(f_1,\ldots \in L^1(\lambda )\) is \(\lambda \)-Cauchy then there exists \(g\in L^1(\lambda )\) so that \(f_1,f_2,\ldots \) converges in measure to \(g\).

Is \(\{v\in \ell ^2(\mathbb {N}): |v_i|\le 1/\log (i+4)\}\) compact in the norm topology on \(\ell ^2(\mathbb {N})\)? Justify your answer.

Define the Fourier transform on \(\mathbb {R}\) (do not worry about the choice of normalization). Show that the Fourier transform of an \(L^1(\mathbb {R},\lambda )\) function is in \(C_0(\mathbb {R})\). You may use that the Fourier transform of a continuous compactly supported function is in \(C_0(\mathbb {R})\).

Let \(V\), \(W\) be Banach spaces, and let \(T_{i} : V \to W\) be a sequence of bounded linear operators such that \(\lim _{i \to \infty } T_{i}v\) exists for every \(v \in V\). Define the linear map \(T : V \to W\) by \[ Tv = \lim _{i \to \infty } T_{i}v. \]

Show that \(T\) is bounded.

Show that the Banach space \(\ell ^{p}\) with the usual norm is not a Hilbert space

when \(p \in [1, 2) \cup (2, \infty )\).

Prove or disprove:

1. \(\ell ^{1}\) is separable.
2. \(\ell ^{\infty }\) is separable.

Let \(C([0,1])\) be the Banach space of continuous functions on \([0,1]\) with the sup norm and view it as a natural subspace of \(L^\infty ([0,1])\), where \([0,1]\) is given Lebesgue measure, and the norm is essential supremum. Prove the existence of a bounded functional on \(L^\infty ([0,1])\) which is not identically \(0\) but vanishes on \(C([0,1])\).

Let \(\mathcal M\) be a \(\sigma \)-algebra of subsets of \(X\) and \(\mu \) a measure on \(\mathcal M\). Let \(A_1\subseteq A_2\subseteq \cdots \) be a sequence of sets in \(\mathcal M\). Let \[ A=\bigcup _{i=1}^{\infty } A_i. \] Prove that \(\lim _{i\to \infty }\mu (A_i)=\mu (A)\).

Now assume that \(\mu (X)<\infty \), and let \(A_1\supseteq A_2\supseteq \cdots \) be a sequence of sets in \(\mathcal M\). Let \[ A=\bigcap _{i=1}^{\infty } A_i. \] Prove that \(\lim _{i\to \infty }\mu (A_i)=\mu (A)\). Give an example where this fails if \(\mu (X)=\infty \).

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots \) be an orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\). Prove that the unit ball \(B\), \(\lVert x\rVert \le 1\), is closed in the weak topology. Prove that the unit sphere \(S\), \(\lVert x\rVert =1\), is dense in \(B\) in the weak topology.

Give an example of a sequence of numbers \(a_1,\ldots \) so that there is no holomorphic function \(f:\mathbb D\to \mathbb C\) so that \(a_1=f'(0)\), \(a_2=f''(0)\), \(a_3=f'''(0)\), \(\ldots \). Justify your answer.

Let \(X=[0,1]\) with the usual measure. For \(n=1,2,\ldots \), let \(f_n=n^{1/p}\cdot \chi _{[0,1/n]}\). Prove that the sequence \(f_n\) does not converge weakly to \(0\) in \(L^1([0,1])\).

Let \((X,\mathcal M,\mu )\) be a measure space. Assume there exists \(\delta >0\) such that for every \(E\in \mathcal M\), either \(\mu (E)=0\) or \(\mu (E)>\delta \). Let \(f\in L^1(X)\). Prove that \(f\) is essentially bounded.

Let \(H=L^2(\mathbb R/\mathbb Z)\), with the usual Lebesgue measure on \(\mathbb R/\mathbb Z\) of total measure \(1\). It is well known that this space has a Hilbert basis \(e_n=e^{2\pi i n x}\), \(n\in \mathbb Z\). Let \(f\in H\) be given by \(f(x)=x\) on \([0,1)\). Compute \(\lVert f\rVert _2^2\) directly and then using Parseval’s equality.

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots \) be an orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\). Prove that the unit ball \(B\), \(\lVert x\rVert \le 1\), is closed in the weak topology. Prove that the unit sphere \(S\), \(\lVert x\rVert =1\), is dense in \(B\) in the weak topology.

Let \(X=[0,1]\) with the usual measure. Let \(p\ge 1\). For \(n=1,2,\ldots \), let \[ f_n=n^{1/p}\cdot \chi _{[0,1/n]}. \] Prove that the sequence \(f_n\) converges weakly to \(0\) in \(L^p([0,1])\) if and only if \(p>1\). Use that \[ L^p([0,1])^*\cong L^q([0,1]). \]

Let \(H\) be a Hilbert space and let \(e_1,e_2,\ldots \) be an orthonormal basis. Prove that \(0\) is a weak limit of the sequence \(e_n\). Prove that the unit ball \(B\), \(\lVert x\rVert \le 1\), is closed in the weak topology. Prove that the unit sphere \(S\), \(\lVert x\rVert =1\), is dense in \(B\) in the weak topology.

Notation: \(D=\{z\in \mathbb C:|z|<1\}\).

Prove that the convolution of two \(L^1(\lambda )\) functions is in \(L^1(\lambda )\).

This question considers different notions of convergence for \(\ell ^1(\mathbb N)\):

(a) Does pointwise convergence imply \(\lVert \cdot \rVert _1\) convergence? Vice versa?

(b) Does pointwise convergence imply convergence in measure? Vice versa?

Show that if \(f:[0,\infty )\to \mathbb C\) is continuous and compactly supported, then \[ \lim _{\rho \to 1}\int |f(x^\rho )-f(x)|\,d\lambda =0. \] Use this to show that if \(f\in L^\infty ([1,2],\lambda )\), we have that \[ \lim _{\rho \to 1^+}\int |f(x^\rho )-f(x)|\,d\lambda =0. \]

Prove the Banach-Steinhaus theorem. That is, show that if \((B,\lVert \cdot \rVert _B)\) and \((\hat B,\lVert \cdot \rVert _{\hat B})\) are Banach spaces and \(A_i:B\to \hat B\) are a sequence of continuous linear maps from \(B\) to \(\hat B\), then either there exists \(M\) so that \(\lVert A_i\rVert _{\mathrm {op}}<M\) for all \(i\), or there exists \(x\in B\) so that \[ \sup _i\lVert A_i x\rVert _{\hat B}=\infty . \]

Consider the interval \([-1,1]\) with a finite Borel measure \(\mu \) (not assumed to be the Lebesgue measure). Let \(C([-1,1])\) be the space of continuous functions endowed with the \(L^2\) norm, and let \(T:C([-1,1])\to \mathbb R\) be the operator \(T(f)=f(0)\). Show that \(T\) is a bounded operator (with respect to the \(L^2\) norm) if and only if \(\mu (\{0\})>0\).

For \(f:\mathbb R\to \mathbb R\) and \(y\in \mathbb R\), define \(\tau _y f(x)=f(x-y)\) for \(x\in \mathbb R\). Assume that \(\mathbb R\) is equipped with the Lebesgue measure. For \(f\in L^1(\mathbb R)\), show that the map \(\mathbb R\to L^1(\mathbb R)\) defined by \(y\mapsto \tau _y f\) is uniformly continuous.

If \(f\in L^1(\lambda )\) then \[ \lim _{c\to 1}\int |f(cx)-f(x)|\,d\lambda =0. \]

Show that if \(f\in L^1\), then \[ \lim _{x\to \infty }\mathcal F(f)(x)=0. \]

Show that \[ \left \{\bar v\in \ell ^p:\ |v_i|\le 2^{-i}\right \} \] is compact for all \(p\ge 1\).

A family of complex-valued functions \(\mathcal S\) on \([0,1]\) is uniformly integrable if for all \(\epsilon >0\) there exists \(\delta >0\) so that \(\lambda (A)<\delta \) implies \(\left |\int _A f\,d\lambda \right |<\epsilon \) for all \(f\in \mathcal S\).

Let \((f_i;i\in \mathbb N)\) be a uniformly integrable sequence. Assume that \[ \lim _{i\to \infty }f_i(x)=g(x)\quad \text {for all }x\in [0,1]. \] Prove that \[ \lim _{i\to \infty }\int _0^1 f_i\,d\lambda =\int _0^1 g\,d\lambda . \]

Let \(f:\mathbb R^2\to \mathbb C\) be a Borel measurable function. Let \(g:\mathbb R^2\to \mathbb R^2\) be the map given by \(g(x,y)=(2x+y^2,y)\) for \((x,y)\in \mathbb R^2\). If \(f\in L^1(\lambda ^2)\), what is \[ \frac {\int _{\mathbb R^2}(f\circ g)\,d\lambda ^2} {\int _{\mathbb R^2}f\,d\lambda ^2}? \]

Consider \(\ell ^4(\mathbb N)\) equipped with topology induced by the norm \(\lVert \cdot \rVert _4\). Show that there exists an open set in this topology that is not open in the weak topology on \(\ell ^4(\mathbb N)\).

(a) If \(f_i\to g\) are in \(L^1(\mathbb R,\lambda )\) and \(\phi \in C(\mathbb R)\), does \[ \lim _{i\to \infty }\int f_i\phi \,d\lambda =\int g\phi \,d\lambda ? \]

(b) Show that if \(f_i\to g\) in \(L^1(\lambda ,[0,1])\), then there exists a subsequence \(n_1,\ldots \) so that \(f_{n_i}\to g\) almost everywhere.

Let \(f\) be in \(L^1(\lambda ,\mathbb R)\). Show that for any \(r>0\) the function \[ h(s)=\int _{s-r}^{s+r} f\,d\lambda \] is continuous.

If \(f_i:\mathbb R\to \mathbb R\) are measurable functions, then is \[ \{x:2\text { is a limit point of }\{f_i(x)\}_{i=1}^{\infty }\} \] measurable?

Recall that \(p\) is a limit point of a sequence \(\{a_i\}_{i=1}^{\infty }\) if there exists an increasing sequence of natural numbers \(n_i\) so that \(p=\lim _{i\to \infty }a_{n_i}\).

Show that if \(f,g\in L^1(\lambda ,\mathbb R)\), then \(\lVert f*g\rVert _1\le \lVert f\rVert _1\lVert g\rVert _1\). Recall \[ (f*g)(x)=\int f(t)g(x-t)\,d\lambda (t). \]

Recall that if \(A\) is a linear operator on Hilbert space with bounded operator norm, then it has an adjoint \(A^*\) which is the unique operator so that for all \(v,w\), \[ \langle Av,w\rangle =\langle v,A^*w\rangle . \]

(a) Let \(P:\ell ^2(\mathbb N)\to \ell ^2(\mathbb N)\) by \(P\bar v=\bar w\) where \(w_j=0\) if \(j\) is not prime and \(w_j=v_j\) otherwise. What is \(P^*\)?

(b) Let \(P:\ell ^2(\mathbb N)\to \ell ^2(\mathbb N)\) by \(P\bar v=\bar w\) where \(w_j=v_{p_j}\), where \(p_j\) is the \(j\)th prime. What is \(P^*\)?

If \(f:\mathbb R\to \mathbb C\) is in \(L^1(\lambda )\), show that the sequence of \(L^1(\lambda )\) functions \[ g_n(x)=f\left (x+\frac {1}{n}\right ) \] converges to \(f\) in \(L^1(\lambda )\).

Consider \(L^2(\mathbb Z):=H\). Recall two topologies on the set of bounded linear operators on \(H\): The norm topology which is metric topology given by \[ d(A,B)=\sup _{v\in H:\lVert v\rVert =1}\lVert Av-Bv\rVert . \] The strong operator topology which is given by \(A_i\to A_\infty \) if for every \(v\in H\) we have \[ \lim _{i\to \infty }\lVert A_iv-A_\infty v\rVert =0. \]

(a) Show that if \(A_i\) converges to \(A_\infty \) in the norm topology then it converges to \(A_\infty \) in the strong operator topology.

(b) Give an example of \(A_i\) which converges to \(A_\infty \) in the strong operator topology but not the norm topology.

(c) Let \(A_i\) be the operator defined by \(A_iv=w\) where \(w_j=v_j\) if \(j\ne i\) and \(w_i=iv_i\). Show \(A_1,\ldots \) is a sequence of operators that does not converge in either topology.

Let \(f_i:[0,1]\to \mathbb R\) be a sequence of functions that converge pointwise to \(f_\infty \).

(a) Show that if the \(f_i\) are uniformly bounded then \[ \lim _{i\to \infty }\int f_i\,d\lambda =\int f_\infty \,d\lambda . \]

(b) What if \(\lVert f_i\rVert _1\le 1\) for all \(i\)?

(c) What if \(\lVert f_i\rVert _2\le 1\) for all \(i\)?

Let \(f\in L^1(\lambda ,[0,1])\) have the property that for any measurable set \(A\) with \(\lambda (A)=\frac {1}{\pi }\) we have \(\int _A f\,d\lambda =0\). Show that \(f=0\) almost everywhere.

Let \((X,M,\mu )\) be a measure space, and suppose that \(1\le p<q<r\). Consider the vector space \(L^p\cap L^r\), with norm \(\lVert f\rVert =\lVert f\rVert _p+\lVert f\rVert _r\). Show that \(L^q\subset L^p\cap L^r\), and that the inclusion map \(i:L^q\to L^p\cap L^r\) is continuous.

Consider \(\mathbb R\) with the Lebesgue measure. Show that \(L^2(\mathbb R)\) is a complete Hilbert space (hint: find a countable, complete orthonormal system).

Let \((X,M,\mu )\) be a \(\sigma \)-finite measure space, and consider \(X\times X\) with the product measure. Suppose that \(K\in L^2(X\times X)\). Show that if \(f\in L^2(X)\), then \(Tf\in L^2(X)\) and \(\lVert Tf\rVert _2\le \lVert K\rVert _2\lVert f\rVert _2\) where \[ Tf(x)=\int _X K(x,y)f(y)\,d\mu (y). \]

Let \(f\in L^p(\mathbb R)\) (with respect to the Lebesgue measure), and \(1\le p\le \infty \), and let \[ F(x)=\int _0^x f(t)\,dt. \] Prove that \(F\) defines a continuous function on \(\mathbb R\).

Let \((X,\mathcal M,\mu )\) be a measure space, \(1<p<\infty \), and \(q=p/(p-1)\). Suppose that \(f_n\in L^p(X)\), \(n=1,2,\ldots \), and that for each \(g\in L^q(X)\), \[ \sup _n \left |\int f_ng\,d\mu \right |<\infty . \] Show that \(\sup _n\lVert f_n\rVert _p<\infty \).

Consider \(\mathbb R\) with the Lebesgue measure, and recall the definitions of convolution and the Fourier transform, for \(f,g\in L^1(\mathbb R)\): \[ f*g(x)=\int f(x-y)g(y)\,dy,\qquad \hat f(\xi )=\int f(x)e^{-2\pi ix\xi }\,dx. \] Show that for \(f,g\in L^1(\mathbb R)\), \(f*g\in L^1(\mathbb R)\), and that \[ \widehat {f*g}=\hat f\,\hat g. \]

Recall that a topological space is said to be separable if it contains a countable dense set. Also recall that \(\ell ^p(\mathbb Z)\) is the \(L^p\)-space for counting measure on the integers. Prove or disprove the following statements:

(a) \(\ell ^1(\mathbb Z)\) is separable.

(b) \(\ell ^\infty (\mathbb Z)\) is separable.

Let \(\mathbb R\) be equipped with Lebesgue measure. Suppose \(f\in L^1(\mathbb R)\) is uniformly continuous. Prove that \(\lim _{|x|\to \infty }f(x)=0\).

Suppose \((X,\mathcal M,\mu )\) is a measure space and fix \(p\) and \(q\) finite such that \[ \frac {1}{p}+\frac {1}{q}=1. \] Let \(f_1,f_2,\ldots \) be a sequence of functions in \(L^p(X)\) converging (in \(L^p\)) to \(f\), and let \(g_1,g_2,\ldots \) be a sequence of functions in \(L^q\) converging (in \(L^q\)) to \(g\). Prove that the sequence \(f_1g_1,f_2g_2,\ldots \) converges to \(fg\) in \(L^1\). Does the same conclusion hold if \(p=1\) and \(q=\infty \)?

Let \(H\) be a Hilbert space and suppose that \(\{x_n\}\) is a sequence in \(H\) with the following property: for each \(y\in H\), \[ \sup _n |\langle x_n,y\rangle |<\infty . \] Prove that \[ \sup _n \lVert x_n\rVert <\infty . \]

Suppose \(1<p<q<r<\infty \). (Here \(p\) and \(q\) are arbitrary, not necessarily conjugate.) Prove that \(L^p(\mathbb R)\cap L^r(\mathbb R)\subset L^q(\mathbb R)\).

Let \([0,1]\) be the unit interval equipped with Lebesgue measure and fix \(1\le p<\infty \). Suppose \(f_1,f_2,\ldots \) is a sequence in \(L^p([0,1])\) which is Cauchy with respect to the \(L^p\) norm.

(a) Prove that there is a subsequence \(g_1,g_2,\ldots \) of \(f_1,f_2,\ldots \) such that \(\lim _{i\to \infty }g_i(x)\) exists for almost all \(x\).

(b) Show (by producing an example of a Cauchy sequence \(f_1,f_2,\ldots \) in \(L^p([0,1])\)) that \(\lim _{i\to \infty }f_i(x)\) can fail to exist for almost all \(x\).

Suppose \((X,\mathcal M,\mu )\) is a measure space and that \(L^p(X)\subset L^q(X)\) for some \(1\le p<q<\infty \).

(a) Show that the inclusion \(L^p(X)\to L^q(X)\) is bounded.

(b) Let \(\mathcal M'\) denote the subsets in \(\mathcal M\) of nonzero measure. Use (a) to prove that \[ \inf _{E'\in \mathcal M'}\mu (E')>0. \]

(a) Let \(\ell ^1\) and \(\ell ^\infty \) be the Banach spaces of sequences of complex numbers \(\xi =(\xi _1,\xi _2,\ldots )\) such that \[ \xi \in \ell ^1 \quad \text {if and only if}\quad \lVert \xi \rVert _1:=\sum _{i=1}^{\infty }|\xi _i|<\infty \] and \[ \xi \in \ell ^\infty \quad \text {if and only if}\quad \lVert \xi \rVert _\infty :=\sup _i|\xi _i|<\infty . \] Let \((\ell ^\infty )^*\) denote the Banach space of continuous linear functionals, equipped with the operator norm, on \(\ell ^\infty \). Show that the map \(\Lambda :\ell ^1\to (\ell ^\infty )^*\) defined by \[ \Lambda (\xi )(\eta _1,\eta _2,\ldots )=\sum _i \xi _i\eta _i \] is a well-defined norm-preserving injection of \(\ell ^1\) into \((\ell ^\infty )^*\) which is not onto.

Hint: Show that \((\ell ^\infty )^*\) contains nontrivial functionals which vanish on the subspace consisting of those \(\xi \) such that \(\xi _i\to 0\) as \(i\to \infty \).

(b) Let \((X,\mu )\) be a measure space. State precise conditions on \(\mu \), \(p\), and \(q\) which guarantee a norm-preserving isomorphism of \(L^p(X)\) onto \((L^q(X))^*\).

Recall that the Fourier transform of a function \(f\in L^1(\mathbb R)\) (with respect to Lebesgue measure) is given by \[ \hat f(x)=\int _{-\infty }^{\infty } e^{-2\pi ixy}f(y)\,dy \] Show that \(\hat f\) is a bounded, continuous function on \(\mathbb R\).

Let \(H\) be a Hilbert Space, and suppose that \(\{x_n\}\) is a sequence in \(H\) such that \(\langle x_n,y\rangle \) is convergent for each \(y\in H\). Prove that there exists \(x\in H\) such that \(\langle x_n,y\rangle \to \langle x,y\rangle \) for all \(y\in H\).

Let \((X,\mu )\) be a finite measure space, and let \(f\in L^\infty (X)\). For \(g\in L^1(X)\), define \[ Tg=\int fg\,d\mu \] Show that \(T\) is a bounded linear functional on \(L^1(X)\), and that \(\lVert T\rVert =\lVert f\rVert _\infty \).

Either prove the following statement if true, or prove that it is false by providing a counterexample. Let \(\{f_n\}\) be a sequence in \(L^2([0,1])\) such that \(\lVert f_n\rVert _2\to 0\). Then \(f_n\to 0\) almost everywhere.

Let \((X,\mu )\) be a measure space with \(\mu (X)=1\), and let \(f\in L^\infty (X)\). Prove that \(\lVert f\rVert _p\to \lVert f\rVert _\infty \) where \(\lVert f\rVert _p\) denotes the \(L^p\) norm of \(f\).

Let \(H\) be a Hilbert space and suppose that \(\{v_n\}\) is a sequence in \(H\) with the following property. For each \(u\in H\) \[ \sup _n |\langle v_n,u\rangle |<\infty \] Prove that \(\sup _n\lVert v_n\rVert <\infty \).

Let \(f\in L^1(\mathbb R)\) such that \[ \int _{-\infty }^{\infty }|x\hat f(x)|\,dx<\infty \] Prove that there is a differentiable function \(g\) on \(\mathbb R\) such that \(f=g\) almost everywhere.

Let \(\ell ^1\) be the Banach space of real sequences \(a=(a_1,a_2,\ldots )\) with norm \[ \lVert a\rVert _1=\sum _{i=1}^{\infty }|a_i|<\infty , \] and \(\ell ^\infty \) the Banach space of real sequences with norm \(\lVert x\rVert _\infty =\sup _i |x_i|\). Let \(a\in \ell ^1\) be a fixed sequence; for any sequence \(x\in \ell ^\infty \), define a new sequence \(T_a(x)\) by \[ \bigl (T_a(x)\bigr )_n=\sum _{i=1}^{n}a_i x_i \]

(a) Show that this defines a bounded operator \(T_a:\ell ^\infty \to \ell ^\infty \).

(b) Show \(\lVert T_a\rVert =\lVert a\rVert _1\), where the left-hand side denotes the norm of the operator \(T_a:\ell ^\infty \to \ell ^\infty \).

Let \((\mathcal M,X,\mu )\) be a finite positive measure space.

(a) Show that \(L^2(X)\subset L^1(X)\), and that the inclusion \(i:L^2(X)\subset L^1(X)\) has norm \(\lVert i\rVert =\sqrt {\mu (X)}\).

(b) Let \(\mathcal M'\subset \mathcal M\) be the subset of measurable sets \(E\) with \(\mu (E)>0\). Show that if \[ L^1(X)=L^2(X), \] then \[ \inf _{E\in \mathcal M'}\mu (E)>0. \]

Let \(f:[0,1]\to \mathbb R_{\ge 0}\) be a measurable integrable function \(f\in L^1([0,1])\) (with respect to the Lebesgue measure). Show that \[ \int _{[0,1]} x f(x)\,dx =\int _{[0,1]}\left (\int _{[y,1]} f(x)\,dx\right )\,dy \]

Let \(K\) be a continuous function on the square \([0,1]\times [0,1]\), and let \([0,1]\) be endowed with the Lebesgue measure. For \(f\in L^2([0,1])\) define \[ Tf(x)=\int _0^1 K(x,y)f(y)\,dy. \] Show that \(Tf\in L^2([0,1])\) and that \(T:L^2\to L^2\) is a bounded operator with \(\lVert T\rVert \le \lVert K\rVert _\infty \).

Let \(f\in L^1([-\pi ,\pi ])\), and let \(c_n\) be the nth Fourier coefficient of \(f\): \[ c_n=\frac {1}{\sqrt {2\pi }}\int _{-\pi }^{\pi } f(x)e^{-inx}\,dx \quad n\in \mathbb Z \] Prove that \(c_n\to 0\) as \(|n|\to \infty \).

Let \(\ell ^\infty \) denote the Banach space of bounded real sequences with norm \(\lVert x\rVert _\infty =\sup _i |x_i|\), and \(\ell ^1\) the Banach space of real sequences with norm \(\lVert x\rVert _1=\sum _i |x_i|<\infty \). For \(x\in \ell ^\infty \) and \(y\in \ell ^1\) define \(Tx(y)=\sum _i x_i y_i\). Prove that \(T:\ell ^\infty \to (\ell ^1)^*\) is a bounded operator that is one to one and onto. (Here \((\ell ^1)^*\) is the dual space of \(\ell ^1\).)

Let \(f\) be a non-constant meromorphic function on \(\mathbb C\). Show that either there exists a sequence \(z_n\to \infty \) with \(f(z_n)\to 0\) as \(n\to \infty \) or there is a \(z\in \mathbb C\) such \(f(z)=0\).

Assume that \(\mathbb R\) has the Lebesgue measure. Let \(f\in L^\infty (\mathbb R)\). For \(g\in L^2(\mathbb R)\) define \[ M(g)=fg \] Prove that \(M:L^2(\mathbb R)\to L^2(\mathbb R)\) is a bounded linear operator, and that \(\lVert M\rVert =\lVert f\rVert _\infty \).

Let \(H\) be a complex Hilbert space, and let \(T:H\to H\) be a bounded self adjoint operator (so \(T=T^*\)).

a. Prove that if \(\lambda \) is an eigenvalue of \(T\), then \(\lambda \) is real.

b. Show that eigenvectors corresponding to different eigenvalues are orthogonal.

c. Assume in addition that \(T\) is compact. Show that if \(\{\lambda _n\}\) is a sequence of distinct eigenvalues with \(\lambda _n\to \lambda \), then \(\lambda =0\).

Let \(f\in L^1([-\pi ,\pi ])\) and for \(n\in \mathbb Z\) let \(c_n\) be the nth Fourier coefficient of \(f\), that is \[ c_n=\frac {1}{\sqrt {2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx \] Prove that \(c_n\to 0\) as \(|n|\to \infty \).

Let \(I=[0,1]\) be the unit interval with the Lebesgue measure, and let \(I\times I\) be endowed with the product measure. Assume that \(K\in L^2(I\times I)\) and \(f\in L^2(I)\). Define \[ g(x)=\int _0^1 K(x,y)f(y)\,dy \] Show that \(g\in L^2(I)\).

Let \(E_1,\ldots ,E_n\) by a finite collection of disjoint measurable subsets of \(X\) and let \(M\) be a subspace of \(L^2(X)\) spanned by the functions \(\{u_1,\ldots ,u_n\}\) where each \(u_i\) is in \(L^2(X)\) and has support contained in \(E_i\). Let \(P:L^2(X)\to \mathbb R\) be the orthogonal projection. Find a function \(h:X\times X\to \mathbb R\) such that \[ Pf(x)=\int _X h(x,y)f(y)\,d\mu (y) \] for all \(f\in L^2(X)\).

The space \(C[0,1]\) of continuous functions on \([0,1]\) is a subspace of all the \(L^p\) spaces, \(L^p[0,1]\) for \(1\le p\le \infty \). Define a linear operator \(T:C[0,1]\to \mathbb R\) by \(Tf=f(1/2)\). Show that \(T\) is a bounded operator for the \(L^p\)-norm if and only if \(p=\infty \). Show that \(T\) extends to a bounded operator on all of \(L^\infty [0,1]\).

In what follows, \(D_R(z_0)\) and \(\overline {D_R}(z_0)\) denote the open and closed discs, respectively, centered at \(z_0\), with radius \(R\).

Suppose \(\Omega \) and \(\Lambda \) are open subsets of the complex plane. Suppose \(\Omega \) is connected and \(\Lambda \) is a simply connected, proper subset of \(\mathbb C\). If \(\{f_n\}\) is a sequence of holomorphic functions from \(\Omega \) to \(\Lambda \), prove that either \(\{f_n\}\) has a subsequence which converges uniformly on compacta to a holomorphic function on \(\Omega \) or \(\{f_n\}\) itself converges to \(\infty \) at every point of \(\Omega \).

Let \(f_1,f_2,\ldots \) be a sequence of real-valued convex functions on \(\mathbb R\). Define their upper-limit via \[ \overline f(x)=\limsup _{n\to \infty } f_n(x); \] and the lower-limit via \[ \underline f(x)=\liminf _{n\to \infty } f_n(x). \] Prove or find a counterexample to each of the following two statements:

(i) \(\overline f\) is convex.

(ii) \(\underline f\) is convex.

Suppose \(X\) consists of two points \(a\) and \(b\). Define \(\mu (\{a\})=1\), \(\mu (\{b\})=\mu (X)=\infty \), and \(\mu (\emptyset )=0\). For this measure \(\mu \), prove or disprove: \(L^\infty (\mu )\) is the dual space of \(L^1(\mu )\).

Let \((\mathcal M,\mu ,X)\) be a positive measure space. Let \(\mathcal M'\) denote the collection of subsets (in \(\mathcal M\)) whose measures are finite. Prove that \[ L^2(X)\subset L^1(X) \] if and only if \[ \sup _{E\in \mathcal M'}\mu (E)<\infty . \]

Let \(H\) be a Hilbert space with inner product \((\cdot ,\cdot )\). Suppose \(\{x_n\}\) is a sequence of elements in \(H\) with the property that the sequence \(\{(x_n,y)\}\) converges for each \(y\in H\). Prove that there is an element \(x\in H\) such that \(\{(x_n,y)\}\) converges to \((x,y)\) for each \(y\in H\).

Construct a sequence of continuous real-valued functions \(f_n\) on \([0,1]\) such that \(0\le f_n\le 1\) and \[ \lim _{n\to \infty }\int _0^1 f_n(x)\,dx=0, \] but such that the sequence \(\{f_n(x)\}\) converges for no \(x\in [0,1]\).

Provide a rigorous computation of the sum of the following series: \[ \sum _{n=1}^{\infty }\frac {1}{n^4}. \] (Hint: one possible approach is to consider the integral of \(f(z)=\frac {\pi }{z^4}\cot (\pi z)\) over an appropriate sequence of contours.)

Let \((X,\mathfrak M,\mu )\) be a positive measure space, and let \(f\in L^1(X)\). Let \(X_n=\{x\in X: |f(x)|>n\}\). Show that \[ \lim _{n\to \infty } n\mu (X_n)=0. \]

Let \(\mu \) be a positive measure on a measure space \((X,\Sigma )\) such that \(\mu (X)=1\). Let \(f\) be a bounded measurable function on \(X\). Show that \(f\in L^p(X)\) for \(1\le p<\infty \) and \(\|f\|_p\le \|f\|_q\) for \(1<p<q<\infty \).

Let \(f_1,f_2,\cdots ,f_n,\cdots \) be a sequence of functions in \(L^1(\mathbb R)\) such that \[ \lim _{n\to \infty }\int _{\mathbb R} f_n(x)g(x)\,dx=0 \] for every \(g\in L^\infty (\mathbb R)\). Show that \(f_1,f_2,\cdots ,f_n,\cdots \) is a bounded sequence in \(L^1(\mathbb R)\).

Let \(f,g\in L^1(\mathbb R)\). Let \(A=f^{-1}(\mathbb C-\{0\})\) and \(B=g^{-1}(\mathbb C-\{0\})\). Show that \[ (f*g)^{-1}(\mathbb C-\{0\})\subset A+B \] where \(A+B=\{x+y:x\in A,y\in B\}\).

Consider \(L^1(T)\) to be the space of periodic functions on \(\mathbb R\) of period \(2\pi \) which are integrable on \([0,2\pi ]\). Let \(M\) be a proper closed translation invariant subspace of \(L^1(T)\).

(a) Prove that there exists a non-zero \(h\in L^\infty (T)\) such that \(f*h(x)=0\) for all \(f\in M\), \[ \text {where } f*h(x)=\frac {1}{2\pi }\int _0^{2\pi } f(x-y)h(y)\,dy; \]

(b) use part (a) to prove there must be an \(n\in \mathbb Z\) such that \(\hat f(n)=0\) for every \(f\in M\), where \(\hat f(n)\) is the \(n\) th Fourier coefficient of \(f\).

Let \(A\) be a closed subspace of \(C([0,1])\) which is also closed in \(L^2([0,1])\). Prove that the orthogonal projection \(P\) which projects \(L^2([0,1])\) onto \(A\) is continuous as a linear transformation from \(L^2([0,1])\) into \(C([0,1])\). Note that \(C([0,1])\) has the topology determined by the norm \(\|\cdot \|_\infty \), while \(L^2([0,1])\) has the topology determined by the norm \(\|\cdot \|_2\).

Let \(\Omega \) be an unbounded domain in the plane with the property that there exists a sequence \(\{h_n\}\) of functions holomorphic on a neighborhood of \(\overline {\Omega }\) such that each \(h_n\) vanishes at infinity on \(\Omega \), \(|h_n|\le 1\) on \(\overline {\Omega }\) for each \(n\), and \(\{h_n\}\) converges uniformly to \(1\) on each compact subset of \(\overline {\Omega }\). Prove that if \(f\) is a function which is bounded and holomorphic in a neighborhood of \(\overline {\Omega }\) and if \(|f|\le 1\) on \(\partial \Omega \), then \(|f|\le 1\) on all of \(\overline {\Omega }\).

Let \(\alpha (x,y)\) be a continuous function on \([0,1]\times [0,1]\). Show that \[ A(f)(x)=\int _0^1 \alpha (x,y)f(y)\,dy \] defines a bounded operator from \(L^2([0,1])\) to \(C([0,1])\), considered as a Banach space with respect to \(\|f\|=\sup _x |f(x)|\).

Consider \[ HC=\{(x_n)\in \mathbb C^{\mathbb Z}\mid |x_n|\le \frac {1}{n}\text { for all }n\in \mathbb Z\}. \] Note that \(HC\) is simply a product of (infinitely many) closed discs. As such, it is compact with respect to the product topology. The product topology on \(HC\) is defined so that sets \[ U_x(m,\delta )=\{(y_n)\in HC\mid |x_n-y_n|<\delta \text { for all }1\le n\le m\} \] provide a fundamental system of neighborhoods of \(x\). Next, note that \(HC\) can be considered a subset of the Hilbert space \(\ell ^2(\mathbb Z)\). Let \(B(x,\epsilon )\) be, as usual, the open ball of radius \(\epsilon \) in \(\ell ^2(\mathbb Z)\). Show that for every \(\epsilon >0\) there exists a positive integer \(m\) and \(\delta >0\) such that \[ U_x(m,\delta )\subseteq B(x,\epsilon )\cap HC \] and, for every positive integer \(m\) and \(\delta >0\) there exists \(\epsilon >0\) such that \[ U_x(m,\delta )\supseteq B(x,\epsilon )\cap HC \] In other words, show that the restricted topology on \(HC\) coincides with the product topology.

Let \(X\) be a measure space with \(\mu (X)=1\). Let \(f\) be a measurable function in \(L^\infty (X)\). Note that \(\|f\|_p\le \|f\|_\infty \) for all \(p\ge 1\). In particular, \(f\) can be considered an element of \(L^p(X)\) for all \(p\ge 1\). Show that \[ \lim _{p\to \infty }\|f\|_p=\|f\|_\infty . \]

Show that the parallelogram law fails for \(L^1([0,1])\) so it is not a Hilbert space.

Hint: Use \(f=\chi _{[0,1/2]}\) and \(g=\chi _{[1/2,1]}\).

Let \(D\) be a domain in the plane, and \(\{f_n\}\) a sequence of functions holomorphic on \(D\). Suppose the sequence is uniformly bounded; that is, there is a positive number \(M\) such that \[ |f_n(z)|<M \quad \text {for all } n \text { and } z\in D. \] Suppose, in addition, that there is an infinite compact set \(K\) in \(D\) such that the sequence \(f_n|K\) converges uniformly on \(K\). Show that the sequence \(f_n\) converges uniformly on all compact subsets of \(D\).

Prove that if \(f,g\in L^2(\mathbb R)\) and the convolution product \(f*g\) is defined by \[ f*g(x)=\int f(y)g(x-y)\,dy, \] then \(f*g(x)\) is a continuous function which vanishes at infinity.

Let \(H_n\) be the linear span in \(L^2([-\pi ,\pi ])\) of the set \(\{e^{ikt}\}_{k=-n}^n\). The orthogonal projection \(P_n\) of \(L^2([-\pi ,\pi ])\) onto \(H_n\) may be written in the form \[ P_n f(x)=h_n*f(x)=\frac {1}{2\pi }\int _{-\pi }^{\pi }h_n(x-t)f(t)\,dt. \] Find the function \(h_n\) and prove that it has this property.

Prove that

(a) \(\log (1+e^t)\le t+\log (2)\) for all \(t\ge 0\);

(b) if \(f\) is a real valued function in \(L^1([0,1])\) then \[ \lim _{n\to \infty }\frac {1}{n}\int _0^1 \log (1+e^{n f(x)})\,dx =\int _0^1 f^+(x)\,dx, \] where \(f^+(x)=f(x)\) if \(f(x)\ge 0\) and \(f^+(x)=0\) if \(f(x)<0\).

Let \(f\) be a continuous positive valued function on \(\mathbb R\), and let \(g\) be the characteristic function of a finite nonempty open interval in \(\mathbb R\). Show that the function \(h=fg\) is in \(L^1(\mathbb R)\), but its Fourier transform \(\hat h\) is not.

Let \(H\) be a Hilbert space with inner product \((\cdot ,\cdot )\) and \(\{x_n\}\) a sequence of elements of \(H\) with the property that the sequence \(\{(x_n,y)\}\) converges for each \(y\in H\). Prove that there is an element \(x\in H\) so that \(\{(x_n,y)\}\) converges to \((x,y)\) for each \(y\in H\).

Let \(f_n\) be a sequence of functions in \(L^\infty (\mathbb R)\) such that \[ \sup _n\left |\int _{\mathbb R} f(x)g(x)\,dx\right |<\infty \] for all \(g\in L^1(\mathbb R)\). Show that \(\sup _n\lVert f_n\rVert <\infty \).

(a) State the definition of a Fourier transform of \(f\in L^1(\mathbb R)\);

(b) State the Fourier Inversion Theorem;

(c) Show that the Fourier transform is one-to-one on \(L^1(\mathbb R)\).

Let \(f(x)=x^{-1/2}\) if \(0<x<1\), \(f(x)=0\) if \(x\le 0\) or \(x\ge 1\), and \(\{r_n\}_{n=1}^{\infty }\) be an enumeration of the rationals. Show that if \[ g(x)=\sum _{n=1}^{\infty }\frac {f(x-r_n)}{2^n} \] then \(g\in L^1(\mathbb R)\) and hence \(g(x)<\infty \) \(m\)-a.e. Show however, that even though \(g^2(x)<\infty \) \(m\)-a.e., that \(g\notin L^2(\mathbb R)\).

Let \(0<p<r<q\). Show that if \((X,\mathfrak M,\mu )\) is a (positive) measure space, then \[ L^r(\mu )\subset L^p(\mu )+L^q(\mu ); \] more precisely, for every \(f\in L^r(\mu )\) there exists \(g\in L^p(\mu )\) and \(h\in L^q(\mu )\) such that \(f=g+h\).

Suppose that \(f\in L^1(\mathbb R^n)\) and \(\hat f\in L^1(\mathbb R^n)\) (\(\hat f\) denoting the Fourier transform of \(f\)). Then show that \[ f\in L^1(\mathbb R^n)\cap L^2(\mathbb R^n). \]

Let \((X,\mathfrak M,\mu )\) be a measure space, and let \(A_1\supset A_2\supset \cdots \supset A_n\supset \cdots \) be a sequence of sets in \(\mathfrak M\).

Is it always true that \(\mu (\bigcap _{n=1}^{\infty }A_n)=\lim _{n\to \infty }\mu (A_n)\)? If so, give a proof. Otherwise give a counterexample.

Let \(\ell ^\infty \) be the normed space consisting of all complex valued, bounded sequences \[ \mathbf a=(a_1,a_2,\cdots a_n\cdots ) \] with the norm \(\lVert \mathbf a\rVert =\sup \{|a_n|:n\in \mathbb N\}\). Let \(W\subset \ell ^\infty \) be the subspace consisting of all \(\mathbf a\in \ell ^\infty \) such that \(\{n\in \mathbb N:a_n\ne 0\}\) is a finite set.

Show that there exists a nontrivial bounded linear functional on \(\ell ^\infty \) which vanishes on \(W\).

Let \(f\in L^1(\mathbb R)\). Define a function \(\varphi \) on \(\mathbb R\) by the formula \(\varphi (x)=\int _{-\infty }^{x}f(t)\,dt\). Show that \(\varphi \) is a uniformly continuous function on \(\mathbb R\).

Using the definitions of convolution and of Fourier transform show that, for \(f,g\in L^1(\mathbb R)\), \[ \widehat {(f*g)}(t)=\hat f(t)\hat g(t) \] for \(t\in \mathbb R\).

Let \(\mu \) be a positive finite measure on \((X,\mathfrak M)\) and \(g\in L^1(\mu )\). Suppose there exists \(0<M<\infty \) such that \[ \left |\int gs\,d\mu \right |\le M\lVert s\rVert _p\qquad (1\le p<\infty ) \] for all simple measurable functions on \(X\). Then show that \(g\in L^q(\mu )\) where \(\frac {1}{p}+\frac {1}{q}=1\).

Let \((X,\mathfrak M,\mu )\) be a positive measure space such that \(\mu (X)=1\). If \(V\) is a closed subspace of \(L^2(\mu )\) which is contained in \(L^\infty (\mu )\) then show that \(\dim (V)<\infty \) as follows:

(1) Show that there exists \(C>0\) such that \(\lVert f\rVert _\infty \le C\lVert f\rVert _2\) for all \(f\in V\).

(2) Let \(\{f_1,\ldots ,f_n\}\) be any orthonormal set in \(V\) and show that for all \(c_1,\ldots ,c_n\in \mathbb C\), and almost all \(x\in X\), \[ \left |\sum _{i=1}^{n}c_i f_i(x)\right | \le C\left (\sum _{i=1}^{n}|c_i|^2\right )^{1/2}. \]

(3) Choose \(c_i=\overline {f_i(x)}\) in (2) and deduce that \[ \sum _{i=1}^{n}|f_i(x)|^2\le C^2\qquad \text {(for almost all }x\in X\text {)}. \]

(4) Integrate the inequality in (3).

Recall the \(k\)th Fourier coefficient of \(f\in L^1([-\pi ,\pi ])\) is \[ \hat f(k)=\frac {1}{2\pi }\int _{-\pi }^{\pi }f(t)e^{-ikt}\,dt. \]

(1) If \(f(t)=t\) for \(-\pi \le t<\pi \), show that \(\hat f(k)=(-1)^{k+1}\frac {1}{ik}\) for \(k\ne 0\) and \(\hat f(0)=0\);

(2) Show that \(\sum _{k=1}^{\infty }k^{-2}=\frac {\pi ^2}{6}\).

Let \((X,\mathfrak M,\mu )\) be a positive measure space. Let \(\psi \in L^2(\mu )\cap L^\infty (\mu )\). Show that \(\psi \in L^p(\mu )\) for all \(p\ge 2\).

Let \(f(z)=\sum _{n=0}^{\infty }a_nz^n\). Assume that \(a_0=0\) and \(a_1=1\). Prove that \(f\) is one to one on the unit disk \(U=\{z\in \mathbb C:|z|<1\}\) if \(\sum _{\ell =2}^{\infty }|a_\ell |\le 1\). (Hint: Show that for every \(z_0\in U\) the function \(z\to f(z)-f(z_0)=z-z_0+\sum _{n=2}^{\infty }a_n(z^n-z_0^n)\) has a simple zero in \(U\).)

Let \(H=L^2(\mathbb R)\) be the Hilbert space with the inner product \(\int _{\mathbb R}f(x)\bar g(x)\,dx\), where \(dx\) is the Lebesgue measure. Let \(T:H\to M\) be the orthogonal projection of \(H\) onto a finite dimensional subspace \(M\) of \(H\).

Show that there exists a function \(h:\mathbb R^2\to \mathbb C\) such that \[ Tf(x)=\int _{\mathbb R}h(x,y)f(y)\,dy \] for all \(f\in L^2(\mathbb R)\).

(Hint: Let \(u_1,\ldots ,u_n\) be an orthonormal basis of \(M\). You may express \(h\) in terms of \(u_1,\ldots ,u_n\).)

Let \(X=[a,b]\subset \mathbb R\) be a closed interval. Assume \(a<b\). Regard \(X\) as a measure space with respect to the usual Lebesgue measure.

(a) Show directly that the space \(C(X)\) of continuous functions on \(X\) is not dense in \(L^\infty (X)\).

(b) Using (a), if necessary, show that there exists a bounded linear functional on \(L^\infty (X)\) which vanishes on \(C(X)\).

Let \(X\) be as in Problem 2. Give an example of a sequence of measurable, nonnegative functions \(f_n:X\to \mathbb R\), \(f_1\ge f_2\ge \cdots \ge f_n\ge \cdots \), converging pointwise to zero, such that \(\int _X f_n(x)\,dx\) does not converge to zero.

Can any of the integrals \(\int _X f_n(x)\,dx\) be finite? Justify your answer.

For \(f:\mathbb R\to \mathbb C\), and \(t\in \mathbb R\) define \[ f_t(x)=f(x-t). \] Suppose \(f\) is continuous with compact support. Prove directly that \(t\to f_t\) is a continuous map of \(\mathbb R\) into \(L^1(\mathbb R)\).

Let \(f\in L^\infty (\mathbb R)\) define a bounded linear operator \(M_f\) from \(L^2(\mathbb R)\) to \(L^2(\mathbb R)\) by \(M_f(g)=fg\).

(a) Prove that \(\lVert M_f\rVert =\lVert f\rVert _\infty \).

(b) Prove that if \(M_f\) is one to one and onto then there exists \(\epsilon >0\) and a set \(E\) of measure zero such that \(|f(x)|\ge \epsilon \) for \(x\in \mathbb R-E\).

Let \((X,\mathcal M,\mu )\) be a measure space, where the measure \(\mu \) is positive. Suppose \(f,f_n\in L^1(X,\mu )\), \(f_n\ge 0\) for each \(n\), \(f_n\to f\) a.e. and \[ \int _X f_n\,d\mu \to \int _X f\,d\mu \] Prove that \(f_n\to f\) in \(L^1(X,\mu )\) norm.

Hint: Show first that \((f-f_n)^+\le f\) for each \(n\) (recall that for real valued function \(g\), \(g^+=\max (g,0)\)).

In \(L^2((0,2\pi ))\) consider the orthogonal projection onto the subspace spanned by \(\{e^{-it},1,e^{it}\}\). Express this projection as an integral operator (i.e. an operator of the form \(f\to Pf\), where \(Pf(y)=\int _0^{2\pi }f(x)h(x,y)\,dx\)).

Let \(H(D)\) be the space of holomorphic functions on an open connected set \(D\subset \mathbb C\). Suppose \(f_n\) is a sequence in \(H(D)\) converging to a function \(f\) uniformly on compact subsets. Suppose also that \(f\) is a nonconstant function. Let \(a\in D\), \(\alpha =f(a)\).

Show there is a sequence \(a_n\in D\) which converges to \(a\), such that \(f_n(a_n)=\alpha \) for each \(n\).