Analysis Ii
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Paper 1, Section II, E
commentLet be an open subset. State what it means for a function to be differentiable at a point , and define its derivative .
State and prove the chain rule for the derivative of , where is a differentiable function.
Let be the vector space of real-valued matrices, and the open subset consisting of all invertible ones. Let be given by .
(a) Show that is differentiable at the identity matrix, and calculate its derivative.
(b) For , let be given by and . Show that on . Hence or otherwise, show that is differentiable at any point of , and calculate for .
Paper 2, Section I, E
commentConsider the map given by
where denotes the unique real cube root of .
(a) At what points is continuously differentiable? Calculate its derivative there.
(b) Show that has a local differentiable inverse near any with .
You should justify your answers, stating accurately any results that you require.
Paper 2, Section II, 12E
comment(a) (i) Define what it means for two norms on a vector space to be Lipschitz equivalent.
(ii) Show that any two norms on a finite-dimensional vector space are Lipschitz equivalent.
(iii) Show that if two norms on a vector space are Lipschitz equivalent then the following holds: for any sequence in is Cauchy with respect to if and only if it is Cauchy with respect to .
(b) Let be the vector space of real sequences such that . Let
and for , let
You may assume that and are well-defined norms on .
(i) Show that is not Lipschitz equivalent to for any .
(ii) Are there any with such that and are Lipschitz equivalent? Justify your answer.
Paper 3, Section I,
comment(a) Let . What does it mean for a function to be uniformly continuous?
(b) Which of the following functions are uniformly continuous? Briefly justify your answers.
(i) on .
(ii) on .
(iii) on .
Paper 3, Section II, E
comment(a) Carefully state the Picard-Lindelöf theorem on solutions to ordinary differential equations.
(b) Let be the set of continuous functions from a closed interval to , and let be a norm on .
(i) Let . Show that for any the norm
is Lipschitz equivalent to the usual sup norm on .
(ii) Assume that is continuous and Lipschitz in the second variable, i.e. there exists such that
for all and all . Define by
for .
Show that there is a choice of such that is a contraction on . Deduce that for any , the differential equation
has a unique solution on with .
Paper 4, Section I, E
commentLet . What does it mean to say that a sequence of real-valued functions on is uniformly convergent?
(i) If a sequence of real-valued functions on converges uniformly to , and each is continuous, must also be continuous?
(ii) Let . Does the sequence converge uniformly on ?
(iii) If a sequence of real-valued functions on converges uniformly to , and each is differentiable, must also be differentiable?
Give a proof or counterexample in each case.
Paper 4, Section II, E
comment(a) (i) Show that a compact metric space must be complete.
(ii) If a metric space is complete and bounded, must it be compact? Give a proof or counterexample.
(b) A metric space is said to be totally bounded if for all , there exists and such that
(i) Show that a compact metric space is totally bounded.
(ii) Show that a complete, totally bounded metric space is compact.
[Hint: If is Cauchy, then there is a subsequence such that
(iii) Consider the space of continuous functions , with the metric
Is this space compact? Justify your answer.
Paper 1, Section II, F
commentLet be a non-empty open set and let .
(a) What does it mean to say that is differentiable? What does it mean to say that is a function?
If is differentiable, show that is continuous.
State the inverse function theorem.
(b) Suppose that is convex, is and that its derivative at a satisfies for all , where is the identity map and denotes the operator norm. Show that is injective.
Explain why is an open subset of .
Must it be true that ? What if ? Give proofs or counter-examples as appropriate.
(c) Find the largest set such that the map given by satisfies for every .
Paper 2, Section I, F
commentShow that defines a norm on the space of continuous functions .
Let be the set of continuous functions with . Show that for each continuous function , there is a sequence with such that as
Show that if is continuous and for every then .
Paper 2, Section II, F
comment(a) Let be a metric space, a non-empty subset of and . Define what it means for to be Lipschitz. If is Lipschitz with Lipschitz constant and if
for each , show that for each and that is Lipschitz with Lipschitz constant . (Be sure to justify that , i.e. that the infimum is finite for every .)
(b) What does it mean to say that two norms on a vector space are Lipschitz equivalent?
Let be an -dimensional real vector space equipped with a norm . Let be a basis for . Show that the map defined by is continuous. Deduce that any two norms on are Lipschitz equivalent.
(c) Prove that for each positive integer and each , there is a constant with the following property: for every polynomial of degree , there is a point such that
where is the derivative of .
Paper 3, Section I,
commentFor a continuous function , define
Show that
for every continuous function , where denotes the Euclidean norm on .
Find all continuous functions with the property that
regardless of the norm on .
[Hint: start by analysing the case when is the Euclidean norm .]
Paper 3, Section II, F
comment(a) Let and let be functions for What does it mean to say that the sequence converges uniformly to on ? What does it mean to say that is uniformly continuous?
(b) Let be a uniformly continuous function. Determine whether each of the following statements is true or false. Give reasons for your answers.
(i) If for each and each , then uniformly on .
(ii) If for each and each , then uniformly on .
(c) Let be a closed, bounded subset of . For each , let be a continuous function such that is a decreasing sequence for each . If is such that for each there is with , show that there is such that .
Deduce the following: If is a continuous function for each such that is a decreasing sequence for each , and if the pointwise limit of is a continuous function , then uniformly on .
Paper 4, Section I, F
commentState the Bolzano-Weierstrass theorem in . Use it to deduce the BolzanoWeierstrass theorem in .
Let be a closed, bounded subset of , and let be a function. Let be the set of points in where is discontinuous. For and , let denote the ball . Prove that for every , there exists such that whenever and .
(If you use the fact that a continuous function on a compact metric space is uniformly continuous, you must prove it.)
Paper 4, Section II, F
comment(a) Define what it means for a metric space to be complete. Give a metric on the interval such that is complete and such that a subset of is open with respect to if and only if it is open with respect to the Euclidean metric on . Be sure to prove that has the required properties.
(b) Let be a complete metric space.
(i) If , show that taken with the subspace metric is complete if and only if is closed in .
(ii) Let and suppose that there is a number such that for every . Show that there is a unique point such that .
Deduce that if is a sequence of points in converging to a point , then there are integers and such that for every .
Paper 1, Section II, G
commentWhat does it mean to say that a real-valued function on a metric space is uniformly continuous? Show that a continuous function on a closed interval in is uniformly continuous.
What does it mean to say that a real-valued function on a metric space is Lipschitz? Show that if a function is Lipschitz then it is uniformly continuous.
Which of the following statements concerning continuous functions are true and which are false? Justify your answers.
(i) If is bounded then is uniformly continuous.
(ii) If is differentiable and is bounded, then is uniformly continuous.
(iii) There exists a sequence of uniformly continuous functions converging pointwise to .
Paper 2, Section I, G
commentLet . What does it mean to say that a sequence of real-valued functions on is uniformly convergent?
Let be functions.
(a) Show that if each is continuous, and converges uniformly on to , then is also continuous.
(b) Suppose that, for every converges uniformly on . Need converge uniformly on ? Justify your answer.
Paper 2, Section II, G
commentLet be a real vector space. What is a norm on ? Show that if is a norm on , then the maps for ) and (for ) are continuous with respect to the norm.
Let be a subset containing 0 . Show that there exists at most one norm on for which is the open unit ball.
Suppose that satisfies the following two properties:
if is a nonzero vector, then the line meets in a set of the form for some ;
if and then .
Show that there exists a norm for which is the open unit ball.
Identify in the following two cases:
(i) for all .
(ii) the interior of the square with vertices .
Let be the set
Is there a norm on for which is the open unit ball? Justify your answer.
Paper 3, Section I, G
commentWhat does it mean to say that a metric space is complete? Which of the following metric spaces are complete? Briefly justify your answers.
(i) with the Euclidean metric.
(ii) with the Euclidean metric.
(iii) The subset
with the metric induced from the Euclidean metric on .
Write down a metric on with respect to which is not complete, justifying your answer.
[You may assume throughout that is complete with respect to the Euclidean metric.
Paper 3, Section II, G
commentWhat is a contraction map on a metric space ? State and prove the contraction mapping theorem.
Let be a complete non-empty metric space. Show that if is a map for which some iterate is a contraction map, then has a unique fixed point. Show that itself need not be a contraction map.
Let be the function
Show that has a unique fixed point.
Paper 4, Section I, G
commentState the chain rule for the composition of two differentiable functions and .
Let be differentiable. For , let . Compute the derivative of . Show that if throughout , then for some function .
Paper 4, Section II, G
commentLet be a nonempty open set. What does it mean to say that a function is differentiable?
Let be a function, where is open. Show that if the first partial derivatives of exist and are continuous on , then is differentiable on .
Let be the function
Determine, with proof, where is differentiable.
Paper 1, Section II, G
commentLet be a metric space.
(a) What does it mean to say that is a Cauchy sequence in ? Show that if is a Cauchy sequence, then it converges if it contains a convergent subsequence.
(b) Let be a Cauchy sequence in .
(i) Show that for every , the sequence converges to some .
(ii) Show that as .
(iii) Let be a subsequence of . If are such that , show that as .
(iv) Show also that for every and ,
(v) Deduce that has a subsequence such that for every and ,
and
(c) Suppose that every closed subset of has the property that every contraction mapping has a fixed point. Prove that is complete.
Paper 2, Section I, G
comment(a) What does it mean to say that the function is differentiable at the point ? Show from your definition that if is differentiable at , then is continuous at .
(b) Suppose that there are functions such that for every ,
Show that is differentiable at if and only if each is differentiable at .
(c) Let be given by
Determine at which points the function is differentiable.
Paper 2, Section II, G
comment(a) What is a norm on a real vector space?
(b) Let be the space of linear maps from to . Show that
defines a norm on , and that if then .
(c) Let be the space of real matrices, identified with in the usual way. Let be the subset
Show that is an open subset of which contains the set .
(d) Let be the map . Show carefully that the series converges on to . Hence or otherwise, show that is twice differentiable at 0 , and compute its first and second derivatives there.
Paper 3, Section I, G
comment(a) Let be a subset of . What does it mean to say that a sequence of functions is uniformly convergent?
(b) Which of the following sequences of functions are uniformly convergent? Justify your answers.
(i)
(ii) .
(iii) , .
(iv) .
Paper 3, Section II, G
commentLet be a metric space.
(a) What does it mean to say that a function is uniformly continuous? What does it mean to say that is Lipschitz? Show that if is Lipschitz then it is uniformly continuous. Show also that if is a Cauchy sequence in , and is uniformly continuous, then the sequence is convergent.
(b) Let be continuous, and be sequentially compact. Show that is uniformly continuous. Is necessarily Lipschitz? Justify your answer.
(c) Let be a dense subset of , and let be a continuous function. Show that there exists at most one continuous function such that for all , . Prove that if is uniformly continuous, then such a function exists, and is uniformly continuous.
[A subset is dense if for any nonempty open subset , the intersection is nonempty.]
Paper 4, Section I, G
comment(a) What does it mean to say that a mapping from a metric space to itself is a contraction?
(b) State carefully the contraction mapping theorem.
(c) Let . By considering the metric space with
or otherwise, show that there exists a unique solution of the system of equations
Paper 4, Section II, G
comment(a) Let be a real vector space. What does it mean to say that two norms on are Lipschitz equivalent? Prove that every norm on is Lipschitz equivalent to the Euclidean norm. Hence or otherwise, show that any linear map from to is continuous.
(b) Let be a linear map between normed real vector spaces. We say that is bounded if there exists a constant such that for all . Show that is bounded if and only if is continuous.
(c) Let denote the space of sequences of real numbers such that is convergent, with the norm . Let be the sequence with and if . Let be the sequence . Show that the subset is linearly independent. Let be the subspace it spans, and consider the linear map defined by
Is continuous? Justify your answer.
Paper 1, Section II, G
commentDefine what it means for a sequence of functions to converge uniformly on to a function .
Let , where are positive constants. Determine all the values of for which converges pointwise on . Determine all the values of for which converges uniformly on .
Let now . Determine whether or not converges uniformly on .
Let be a continuous function. Show that the sequence is uniformly convergent on if and only if .
[If you use any theorems about uniform convergence, you should prove these.]
Paper 2, Section I, G
commentShow that the map given by
is differentiable everywhere and find its derivative.
Stating accurately any theorem that you require, show that has a differentiable local inverse at a point if and only if
Paper 2, Section II, G
commentLet be normed spaces with norms . Show that for a map and , the following two statements are equivalent:
(i) For every given there exists such that whenever
(ii) for each sequence .
We say that is continuous at if (i), or equivalently (ii), holds.
Let now be a normed space. Let be a non-empty closed subset and define . Show that
In the case when with the standard Euclidean norm, show that there exists such that .
Let be two disjoint closed sets in . Must there exist disjoint open sets such that and ? Must there exist and such that for all and ? For each answer, give a proof or counterexample as appropriate.
Paper 3, Section I, G
commentDefine what is meant by a uniformly continuous function on a subset of a metric space. Show that every continuous function on a closed, bounded interval is uniformly continuous. [You may assume the Bolzano-Weierstrass theorem.]
Suppose that a function is continuous and tends to a finite limit at . Is necessarily uniformly continuous on Give a proof or a counterexample as appropriate.
Paper 3, Section II, G
commentDefine what it means for a function to be differentiable at with derivative .
State and prove the chain rule for the derivative of , where is a differentiable function.
Now let be a differentiable function and let where is a constant. Show that is differentiable and find its derivative in terms of the partial derivatives of . Show that if holds everywhere in , then for some differentiable function
Paper 4, Section I, G
commentDefine what is meant for two norms on a vector space to be Lipschitz equivalent.
Let denote the vector space of continuous functions with continuous first derivatives and such that for in some neighbourhood of the end-points and 1 . Which of the following four functions define norms on (give a brief explanation)?
Among those that define norms, which pairs are Lipschitz equivalent? Justify your answer.
Paper 4, Section II, G
commentConsider the space of bounded real sequences with the norm . Show that for every bounded sequence in there is a subsequence which converges in every coordinate, i.e. the sequence of real numbers converges for each . Does every bounded sequence in have a convergent subsequence? Justify your answer.
Let be the subspace of real sequences such that converges. Is complete in the norm (restricted from to ? Justify your answer.
Suppose that is a real sequence such that, for every , the series converges. Show that
Suppose now that is a real sequence such that, for every , the series converges. Show that
Paper 1, Section II, F
commentDefine what it means for two norms on a real vector space to be Lipschitz equivalent. Show that if two norms on are Lipschitz equivalent and , then is closed in one norm if and only if is closed in the other norm.
Show that if is finite-dimensional, then any two norms on are Lipschitz equivalent.
Show that is a norm on the space of continuous realvalued functions on . Is the set closed in the norm ?
Determine whether or not the norm is Lipschitz equivalent to the uniform on .
[You may assume the Bolzano-Weierstrass theorem for sequences in .]
Paper 2, Section I, F
commentDefine what is meant by a uniformly continuous function on a set .
If and are uniformly continuous functions on , is the (pointwise) product necessarily uniformly continuous on ?
Is a uniformly continuous function on necessarily bounded?
Is uniformly continuous on
Justify your answers.
Paper 2, Section II,
commentLet be subsets of and define . For each of the following statements give a proof or a counterexample (with justification) as appropriate.
(i) If each of is bounded and closed, then is bounded and closed.
(ii) If is bounded and closed and is closed, then is closed.
(iii) If are both closed, then is closed.
(iv) If is open and is closed, then is open.
[The Bolzano-Weierstrass theorem in may be assumed without proof.]
Paper 3, Section I, F
commentLet be an open set and let be a differentiable function on such that for some constant and all , where denotes the operator norm of the linear map . Let be a straight-line segment contained in . Prove that , where denotes the Euclidean norm on .
Prove that if is an open ball and for each , then is constant on .
Paper 3, Section II, F
commentLet , be continuous functions on an open interval . Prove that if the sequence converges to uniformly on then the function is continuous on .
If instead is only known to converge pointwise to and is continuous, must be uniformly convergent? Justify your answer.
Suppose that a function has a continuous derivative on and let
Stating clearly any standard results that you require, show that the functions converge uniformly to on each interval .
Paper 4, Section I, F
commentDefine a contraction mapping and state the contraction mapping theorem.
Let be the space of continuous real-valued functions on endowed with the uniform norm. Show that the map defined by
is not a contraction mapping, but that is.
Paper 4, Section II, F
commentLet be an open set. Define what it means for a function to be differentiable at a point .
Prove that if the partial derivatives and exist on and are continuous at , then is differentiable at .
If is differentiable on must be continuous at Give a proof or counterexample as appropriate.
The function is defined by
Determine all the points at which is differentiable.
Paper 1, Section II, F
commentDefine what it means for a sequence of functions , to converge uniformly on an interval .
By considering the functions , or otherwise, show that uniform convergence of a sequence of differentiable functions does not imply uniform convergence of their derivatives.
Now suppose is continuously differentiable on for each , that converges as for some , and moreover that the derivatives converge uniformly on . Prove that converges to a continuously differentiable function on , and that
Hence, or otherwise, prove that the function
is continuously differentiable on .
Paper 2, Section I, F
commentLet denote the vector space of continuous real-valued functions on the interval , and let denote the subspace of continuously differentiable functions.
Show that defines a norm on . Show furthermore that the map takes the closed unit ball to a bounded subset of .
If instead we had used the norm restricted from to , would take the closed unit ball to a bounded subset of ? Justify your answer.
Paper 2, Section II, F
Let be continuous on an open set . Suppose that on the partial derivatives and exist and are continuous. Prove that