Linear Analysis
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Paper 1, Section II, 22H
commentLet be a separable Hilbert space and be a Hilbertian (orthonormal) basis of . Given a sequence of elements of and , we say that weakly converges to , denoted , if .
(a) Given a sequence of elements of , prove that the following two statements are equivalent:
(i) such that ;
(ii) the sequence is bounded in and , the sequence is convergent.
(b) Let be a bounded sequence of elements of . Show that there exists and a subsequence such that in .
(c) Let be a sequence of elements of and be such that . Show that the following three statements are equivalent:
(i) ;
(ii) ;
(iii) such that .
Paper 2, Section II, 22H
comment(a) Let be a real normed vector space. Show that any proper subspace of has empty interior.
Assuming to be infinite-dimensional and complete, prove that any algebraic basis of is uncountable. [The Baire category theorem can be used if stated properly.] Deduce that the vector space of polynomials with real coefficients cannot be equipped with a complete norm, i.e. a norm that makes it complete.
(b) Suppose that and are norms on a vector space such that and are both complete. Prove that if there exists such that for all , then there exists such that for all . Is this still true without the assumption that and are both complete? Justify your answer.
(c) Let be a real normed vector space (not necessarily complete) and be the set of linear continuous forms . Let be a sequence in such that for all . Prove that
Paper 3, Section II, H
comment(a) State the Arzela-Ascoli theorem, including the definition of equicontinuity.
(b) Consider a sequence of continuous real-valued functions on such that for all is bounded and the sequence is equicontinuous at . Prove that there exists and a subsequence such that uniformly on any closed bounded interval.
(c) Let be a Hausdorff compact topological space, and the real-valued continuous functions on . Let be a compact subset of . Prove that the collection of functions is equicontinuous.
(d) We say that a Hausdorff topological space is locally compact if every point has a compact neighbourhood. Let be such a space, compact and open such that . Prove that there exists continuous with compact support contained in and equal to 1 on . [Hint: Construct an open set such that and is compact, and use Urysohn's lemma to construct a function in and then extend it by zero.]
Paper 4, Section II, H
comment(a) Let be two Hilbert spaces, and be a bounded linear operator. Show that there exists a unique bounded linear operator such that
(b) Let be a separable Hilbert space. We say that a sequence is a frame of if there exists such that
State briefly why such a frame exists. From now on, let be a frame of . Show that is dense in .
(c) Show that the linear map given by is bounded and compute its adjoint .
(d) Assume now that is a Hilbertian (orthonormal) basis of and let . Show that the Hilbert cube such that is a compact subset of .
Paper 1, Section II, I
comment(a) Define the dual space of a (real) normed space . Define what it means for two normed spaces to be isometrically isomorphic. Prove that is isometrically isomorphic to .
(b) Let . [In this question, you may use without proof the fact that is isometrically isomorphic to where .]
(i) Show that if is a countable dense subset of , then the function
defines a metric on the closed unit ball . Show further that for a sequence of elements , we have
Deduce that is a compact metric space.
(ii) Give an example to show that for a sequence of elements and ,
Paper 2, Section II, I
comment(a) State and prove the Baire Category theorem.
Let . Apply the Baire Category theorem to show that . Give an explicit element of .
(b) Use the Baire Category theorem to prove that contains a function which is nowhere differentiable.
(c) Let be a real Banach space. Verify that the map sending to the function is a continuous linear map of into where denotes the dual space of . Taking for granted the fact that this map is an isometry regardless of the norm on , prove that if is another norm on the vector space which is not equivalent to , then there is a linear function which is continuous with respect to one of the two norms and not continuous with respect to the other.
Paper 3, Section II, I
commentLet be a separable complex Hilbert space.
(a) For an operator , define the spectrum and point spectrum. Define what it means for to be: (i) a compact operator; (ii) a self-adjoint operator and (iii) a finite rank operator.
(b) Suppose is compact. Prove that given any , there exists a finite-dimensional subspace such that for each , where is an orthonormal basis for and denotes the orthogonal projection onto . Deduce that a compact operator is the operator norm limit of finite rank operators.
(c) Suppose that has finite rank and is not an eigenvalue of . Prove that is surjective. [You may wish to consider the action of on
(d) Suppose is compact and is not an eigenvalue of . Prove that the image of is dense in .
Prove also that is bounded below, i.e. prove also that there exists a constant such that for all . Deduce that is surjective.
Paper 4, Section II, I
comment(a) For a compact Hausdorff space, what does it mean to say that a set is equicontinuous. State and prove the Arzelà-Ascoli theorem.
(b) Suppose is a compact Hausdorff space for which is a countable union of equicontinuous sets. Prove that is finite.
(c) Let be a bounded, continuous function and let . Consider the problem of finding a differentiable function with
for all . For each , let be defined by setting and
for , where
for and .
(i) Verify that is well-defined and continuous on for each .
(ii) Prove that there exists a differentiable function solving (*) for .
Paper 1, Section II, H
commentLet be the space of real-valued sequences with only finitely many nonzero terms.
(a) For any , show that is dense in . Is dense in Justify your answer.
(b) Let , and let be an operator that is bounded in the -norm, i.e., there exists a such that for all . Show that there is a unique bounded operator satisfying , and that .
(c) For each and for each determine if there is a bounded operator from to (in the norm) whose restriction to is given by :
(d) Let be a normed vector space such that the closed unit ball is compact. Prove that is finite dimensional.
Paper 2, Section II, H
comment(a) State the real version of the Stone-Weierstrass theorem and state the UrysohnTietze extension theorem.
(b) In this part, you may assume that there is a sequence of polynomials such that as .
Let be a continuous piecewise linear function which is linear on and on . Using the polynomials mentioned above (but not assuming any form of the Stone-Weierstrass theorem), prove that there are polynomials such that as .
(d) Which of the following families of functions are relatively compact in with the supremum norm? Justify your answer.
[In this question denotes the set of positive integers.]
Paper 3, Section II, H
comment(a) Let be a Banach space and consider the open unit ball . Let be a bounded operator. Prove that .
(b) Let be the vector space of all polynomials in one variable with real coefficients. Let be any norm on . Show that is not complete.
(c) Let be entire, and assume that for every there is such that where is the -th derivative of . Prove that is a polynomial.
[You may use that an entire function vanishing on an open subset of must vanish everywhere.]
(d) A Banach space is said to be uniformly convex if for every there is such that for all such that and , one has . Prove that is uniformly convex.
Paper 4, Section II, H
comment(a) State and prove the Riesz representation theorem for a real Hilbert space .
[You may use that if is a real Hilbert space and is a closed subspace, then
(b) Let be a real Hilbert space and a bounded linear operator. Show that is invertible if and only if both and are bounded below. [Recall that an operator is bounded below if there is such that for all .]
(c) Consider the complex Hilbert space of two-sided sequences,
with norm . Define by . Show that is unitary and find the point spectrum and the approximate point spectrum of .
Paper 1, Section II, F
commentLet be a compact Hausdorff space.
(a) State the Arzelà-Ascoli theorem, and state both the real and complex versions of the Stone-Weierstraß theorem. Give an example of a compact space and a bounded set of functions in that is not relatively compact.
(b) Let be continuous. Show that there exists a sequence of polynomials in variables such that
Characterize the set of continuous functions for which there exists a sequence of polynomials such that uniformly on .
(c) Prove that if is equicontinuous then is finite. Does this implication remain true if we drop the requirement that be compact? Justify your answer.
Paper 2, Section II, F
commentLet be Banach spaces and let denote the space of bounded linear operators .
(a) Define what it means for a bounded linear operator to be compact. Let be linear operators with finite rank, i.e., is finite-dimensional. Assume that the sequence converges to in . Show that is compact.
(b) Let be compact. Show that the dual map is compact. [Hint: You may use the Arzelà-Ascoli theorem.]
(c) Let be a Hilbert space and let be a compact operator. Let be an infinite sequence of eigenvalues of with eigenvectors . Assume that the eigenvectors are orthogonal to each other. Show that .
Paper 3, Section II, F
comment(a) Let be a normed vector space and let be a Banach space. Show that the space of bounded linear operators is a Banach space.
(b) Let and be Banach spaces, and let be a dense linear subspace. Prove that a bounded linear map can be extended uniquely to a bounded linear map with the same operator norm. Is the claim also true if one of and is not complete?
(c) Let be a normed vector space. Let be a sequence in such that
Prove that there is a constant such that
Paper 4, Section II, F
comment(a) Let be a separable normed space. For any sequence with for all , show that there is and a subsequence such that for all as . [You may use without proof the fact that is complete and that any bounded linear map , where is a dense linear subspace, can be extended uniquely to an element .]
(b) Let be a Hilbert space and a unitary map. Let
Prove that and are orthogonal, , and that for every ,
where is the orthogonal projection onto the closed subspace .
(c) Let be a linear map, where is the unit circle, induced by a homeomorphism by . Prove that there exists with such that for all . (Here denotes the function on which returns 1 identically.) If is not the identity map, does it follow that as above is necessarily unique? Justify your answer.
Paper 1, Section II, F
commentLet be a normed vector space over the real numbers.
(a) Define the dual space of and prove that is a Banach space. [You may use without proof that is a vector space.]
(b) The Hahn-Banach theorem states the following. Let be a real vector space, and let be sublinear, i.e., and for all and all . Let be a linear subspace, and let be linear and satisfy for all . Then there exists a linear functional such that for all and .
Using the Hahn-Banach theorem, prove that for any non-zero there exists such that and .
(c) Show that can be embedded isometrically into a Banach space, i.e. find a Banach space and a linear map with for all .
Paper 2, Section II, F
comment(a) Let be a normed vector space and a closed subspace with . Show that is nowhere dense in .
(b) State any version of the Baire Category theorem.
(c) Let be an infinite-dimensional Banach space. Show that cannot have a countable algebraic basis, i.e. there is no countable subset such that every can be written as a finite linear combination of elements of .
Paper 3, Section II, F
commentLet be a non-empty compact Hausdorff space and let be the space of real-valued continuous functions on .
(i) State the real version of the Stone-Weierstrass theorem.
(ii) Let be a closed subalgebra of . Prove that and implies that where the function is defined by . [You may use without proof that implies .]
(iii) Prove that is normal and state Urysohn's Lemma.
(iv) For any , define by for . Justifying your answer carefully, find
Paper 4, Section II, F
commentLet be a complex Hilbert space with inner product and let be a bounded linear map.
(i) Define the spectrum , the point spectrum , the continuous spectrum , and the residual spectrum .
(ii) Show that is self-adjoint and that . Show that if is compact then so is .
(iii) Assume that is compact. Prove that has a singular value decomposition: for or , there exist orthonormal systems and and such that, for any ,
[You may use the spectral theorem for compact self-adjoint linear operators.]
Paper 1, Section II, I
comment(a) State the closed graph theorem.
(b) Prove the closed graph theorem assuming the inverse mapping theorem.
(c) Let be Banach spaces and be linear maps. Suppose that is bounded and is both bounded and injective. Show that is bounded.
Paper 2, Section II, I
comment(a) Let be a topological space and let denote the normed vector space of bounded continuous real-valued functions on with the norm . Define the terms uniformly bounded, equicontinuous and relatively compact as applied to subsets .
(b) The Arzela-Ascoli theorem [which you need not prove] states in particular that if is compact and is uniformly bounded and equicontinuous, then is relatively compact. Show by examples that each of the compactness of , uniform boundedness of , and equicontinuity of are necessary conditions for this conclusion.
(c) Let be a topological space. Assume that there exists a sequence of compact subsets of such that and . Suppose is uniformly bounded and equicontinuous and moreover satisfies the condition that, for every , there exists such that for every and for every . Show that is relatively compact.
Paper 3, Section II, I
comment(a) Define Banach spaces and Euclidean spaces over . [You may assume the definitions of vector spaces and inner products.]
(b) Let be the space of sequences of real numbers with finitely many non-zero entries. Does there exist a norm on such that is a Banach space? Does there exist a norm such that is Euclidean? Justify your answers.
(c) Let be a normed vector space over satisfying the parallelogram law
for all . Show that is an inner product on . [You may use without proof the fact that the vector space operations and are continuous with respect to . To verify the identity , you may find it helpful to consider the parallelogram law for the pairs and
(d) Let be an incomplete normed vector space over which is not a Euclidean space, and let be its dual space with the dual norm. Is a Banach space? Is it a Euclidean space? Justify your answers.
Paper 4, Section II, I
commentLet be a complex Hilbert space.
(a) Let be a bounded linear map. Show that the spectrum of is a subset of .
(b) Let be a bounded self-adjoint linear map. For , let and . If , show that .
(c) Let be a compact self-adjoint linear map. For , show that is finite-dimensional.
(d) Let be a closed, proper, non-trivial subspace. Let be the orthogonal projection to .
(i) Prove that is self-adjoint.
(ii) Determine the spectrum and the point spectrum of .
(iii) Find a necessary and sufficient condition on for to be compact.
Paper 1, Section II, G
comment(a) Let be an orthonormal basis of an inner product space . Show that for all there is a unique sequence of scalars such that .
Assume now that is a Hilbert space and that is another orthonormal basis of . Prove that there is a unique bounded linear map such that for all . Prove that this map is unitary.
(b) Let with . Show that no subspace of is isomorphic to . [Hint: Apply the generalized parallelogram law to suitable vectors.]
Paper 2, Section II, G
comment(a) Let be a linear map between normed spaces. What does it mean to say that is bounded? Show that is bounded if and only if is continuous. Define the operator norm of and show that the set of all bounded, linear maps from to is a normed space in the operator norm.
(b) For each of the following linear maps , determine if is bounded. When is bounded, compute its operator norm and establish whether is compact. Justify your answers. Here for and for .
(i) .
(ii) .
(iii) .
(iv) , where is a given element of . [Hint: Consider first the case that for every , and apply to a suitable function. In the general case apply to a suitable sequence of functions.]
Paper 3, Section II, G
commentState and prove the Baire Category Theorem. [Choose any version you like.]
An isometry from a metric space to another metric space is a function such that for all . Prove that there exists no isometry from the Euclidean plane to the Banach space of sequences converging to 0 . [Hint: Assume is an isometry. For and let denote the coordinate of . Consider the sets consisting of all pairs with and .]
Show that for each there is a linear isometry .
Paper 4, Section II, G
commentLet be a Hilbert space and . Define what is meant by an adjoint of and prove that it exists, it is linear and bounded, and that it is unique. [You may use the Riesz Representation Theorem without proof.]
What does it mean to say that is a normal operator? Give an example of a bounded linear map on that is not normal.
Show that is normal if and only if for all .
Prove that if is normal, then , that is, that every element of the spectrum of is an approximate eigenvalue of .
Paper 1, Section II, G
commentLet and be normed spaces. What is an isomorphism between and ? Show that a bounded linear map is an isomorphism if and only if is surjective and there is a constant such that for all . Show that if there is an isomorphism and is complete, then is complete.
Show that two normed spaces of the same finite dimension are isomorphic. [You may assume without proof that any two norms on a finite-dimensional space are equivalent.] Briefly explain why this implies that every finite-dimensional space is complete, and every closed and bounded subset of a finite-dimensional space is compact.
Let and be subspaces of a normed space with . Assume that is closed in and is finite-dimensional. Prove that is closed in . [Hint: First show that the function restricted to the unit sphere of F achieves its minimum.]
Paper 2, Section II, G
comment(a) Let and be Banach spaces, and let be a surjective linear map. Assume that there is a constant such that for all . Show that is continuous. [You may use any standard result from general Banach space theory provided you clearly state it.] Give an example to show that the assumption that and are complete is necessary.
(b) Let be a closed subset of a Banach space such that
(i) for each ;
(ii) for each and ;
(iii) for each , there exist such that .
Prove that, for some , the unit ball of is contained in the closure of the set
[You may use without proof any version of the Baire Category Theorem.] Deduce that, for some , every can be written as with and
Paper 3, Section II, G
comment(i) State carefully the theorems of Stone-Weierstrass and Arzelá-Ascoli (work with real scalars only).
(ii) Let denote the family of functions on of the form
where the are real and for all . Prove that any sequence in has a subsequence that converges uniformly on .
(iii) Let be a continuous function such that and exists. Show that for each there exists a real polynomial having only odd powers, i.e. of the form
such that . Show that the same holds without the assumption that is differentiable at 0 .
Paper 4, Section II, G
commentDefine the spectrum and the approximate point spectrum of a bounded linear operator on a Banach space. Prove that and that is a closed and bounded subset of . [You may assume without proof that the set of invertible operators is open.]
Let be a hermitian operator on a non-zero Hilbert space. Prove that is not empty
Let be a non-empty, compact subset of . Show that there is a bounded linear operator with [You may assume without proof that a compact metric space is separable.]
Paper 1, Section II, F
commentState and prove the Closed Graph Theorem. [You may assume any version of the Baire Category Theorem provided it is clearly stated. If you use any other result from the course, then you must prove it.]
Let be a closed subspace of such that is also a subset of . Show that the left-shift , given by , is bounded when is equipped with the sup-norm.
Paper 2, Section II, F
commentLet be a Banach space. Let be a bounded linear operator. Show that there is a bounded sequence in such that for all .
Fix . Define the Banach space and briefly explain why it is separable. Show that for there exists such that and . [You may use Hölder's inequality without proof.]
Deduce that embeds isometrically into .
Paper 3, Section II, F
commentState the Stone-Weierstrass Theorem for real-valued functions.
State Riesz's Lemma.
Let be a compact, Hausdorff space and let be a subalgebra of separating the points of and containing the constant functions. Fix two disjoint, non-empty, closed subsets and of .
(i) If show that there exists such that on , and on . Explain briefly why there is such that on .
(ii) Show that there is an open cover of , elements of and positive integers such that
for each .
(iii) Using the inequality
show that for sufficiently large positive integers , the element
of satisfies
for each .
(iv) Show that the element of satisfies
Now let with . By considering the sets and , show that there exists such that . Deduce that is dense in .
Paper 4, Section II, F
commentLet be a bounded linear operator on a complex Banach space . Define the spectrum of . What is an approximate eigenvalue of ? What does it mean to say that is compact?
Assume now that is compact. Show that if is in the boundary of and , then is an eigenvalue of . [You may use without proof the result that every in the boundary of is an approximate eigenvalue of .]
Let be a compact Hermitian operator on a complex Hilbert space . Prove the following:
(a) If and , then is an eigenvalue of .
(b) is countable.
Paper 1, Section II, G
What is meant by the dual of a normed space ? Show that is a Banach space.
Let , the space of functions possessing a bounded, continuous first derivative. Endow with the sup norm . Which of the following maps are elements of ? Give brief justifications or counterexamples as appropriate.