METRICS, NORMS, INNER
PRODUCTS, AND
OPERATOR THEORY –
SOLUTIONS TO PROBLEMS
(2021 EDITION) – HEIL
Metrics, Norms, Inner Products, and Operator Theory
— Comprehensive Practice Exam
Section 1: Preliminaries & Notation (Questions 1–15)
1. In Heil's textbook, the field F over which vector spaces are defined denotes:
A. Only the real numbers R
B. Only the complex numbers C
C. Either R or C
D. The rational numbers Q
Rationale: Heil explicitly states that F denotes either R or C throughout the text. This
allows the theorems to be presented in both real and complex settings, which is
important for applications in harmonic analysis and quantum mechanics.
,2. The set R^d is the same as:
A. {f : {1, 2, ..., d} → R}
B. {f : R → {1, 2, ..., d}}
C. The set of all polynomials of degree ≤ d
D. The set of all d × d matrices
Rationale: This is a conceptual exercise from the text. A vector (x₁, ..., x_d) in R^d is
equivalently a function from the index set {1, ..., d} to R.
3. The Baire Category Theorem and the Uniform Boundedness Principle are
covered in:
A. Chapter 2 (Metric Spaces)
B. Chapter 3 (Norms and Banach Spaces)
C. Chapter 4 (Further Results on Banach Spaces)
D. Chapter 5 (Inner Products and Hilbert Spaces)
Rationale: The table of contents confirms that Chapter 4 covers "Further Results on
Banach Spaces," which includes these advanced topics.
4. Heil's textbook covers the Spectral Theorem for:
A. All bounded linear operators on Hilbert spaces
B. Compact self-adjoint operators on Hilbert spaces
C. Unitary operators on Banach spaces
D. Normal operators on finite-dimensional spaces
Rationale: The Spectral Theorem for compact self-adjoint operators is covered in
Chapter 7. The general spectral theorem for bounded normal operators requires more
advanced machinery.
5. According to Heil, the three main families of spaces encountered in analysis are:
A. Vector spaces, topological spaces, and measure spaces
B. Metric spaces, normed spaces, and inner product spaces
C. Banach spaces, Hilbert spaces, and Sobolev spaces
D. Finite-dimensional spaces, separable spaces, and reflexive spaces
Rationale: This is stated explicitly in the preface: the text is "a self-contained
introduction to these spaces" — metric, normed, and inner product spaces.
6. The dual of ℓ^p (for 1 < p < ∞) is:
,A. ℓ^q where 1/p + 1/q = 1
B. ℓ^p itself
C. ℓ^∞
D. c₀
Rationale: This is a standard result covered in Chapter 4. The dual of ℓ^p is isometrically
isomorphic to ℓ^q for 1 < p < ∞.
7. A Schauder basis for a Banach space is:
A. A Hamel basis (every vector is a finite linear combination)
B. A sequence such that every vector has a unique convergent series
representation
C. Any linearly independent set that spans the space
D. A maximal orthonormal set
Rationale: Unlike a Hamel basis, a Schauder basis allows infinite series representations.
This is a key topic in Chapter 4 that is "rarely presented in an accessible way to
undergraduate students".
8. The text requires knowledge of measure theory:
A. Throughout all chapters
B. In Chapters 1–7
C. No — measure theory is not required for the main text
D. Only in Chapter 6
Rationale: The text is designed so that "no familiarity with measure theory is required."
An optional online Chapter 8 (Integral Operators) covers related results that do depend
on Lebesgue measure.
9. Chapter 5 covers:
A. Norms and Banach Spaces
B. Metric Spaces
C. Inner Products and Hilbert Spaces
D. Operator Theory
Rationale: The table of contents confirms: Chapter 5 is "Inner Products and Hilbert
Spaces".
10. The online Extra Chapter 8 covers:
, A. Spectral Theory
B. The Uniform Boundedness Principle
C. Integral Operators
D. Fourier Series
Rationale: Chapter 8, "Integral Operators," is posted online and requires measure
theory.
11. Heil recommends a course covering all topics in:
A. One semester
B. Two semesters
C. Three quarters
D. One year
Rationale: The author recommends a 2-semester course if all topics are to be covered.
12. The notation L(X, Y) denotes:
A. The set of all linear transformations from X to Y
B. The set of all bounded linear operators from X to Y
C. The set of all continuous functions from X to Y
D. The Lebesgue space of functions from X to Y
Rationale: L(X, Y) denotes the space of bounded linear operators, which is itself a
normed space when equipped with the operator norm.
13. The dual space X of a normed space X is:*
A. The set of all linear functionals on X
B. The set of all bounded (continuous) linear functionals on X
C. The set of all bounded linear operators from X to X
D. The completion of X
Rationale: X* = L(X, F), the space of bounded linear functionals. It is always a Banach
space.
14. Which theorem states that a normed space X is a Banach space if and only if
every absolutely convergent series converges?
A. The Baire Category Theorem
B. The Uniform Boundedness Principle
PRODUCTS, AND
OPERATOR THEORY –
SOLUTIONS TO PROBLEMS
(2021 EDITION) – HEIL
Metrics, Norms, Inner Products, and Operator Theory
— Comprehensive Practice Exam
Section 1: Preliminaries & Notation (Questions 1–15)
1. In Heil's textbook, the field F over which vector spaces are defined denotes:
A. Only the real numbers R
B. Only the complex numbers C
C. Either R or C
D. The rational numbers Q
Rationale: Heil explicitly states that F denotes either R or C throughout the text. This
allows the theorems to be presented in both real and complex settings, which is
important for applications in harmonic analysis and quantum mechanics.
,2. The set R^d is the same as:
A. {f : {1, 2, ..., d} → R}
B. {f : R → {1, 2, ..., d}}
C. The set of all polynomials of degree ≤ d
D. The set of all d × d matrices
Rationale: This is a conceptual exercise from the text. A vector (x₁, ..., x_d) in R^d is
equivalently a function from the index set {1, ..., d} to R.
3. The Baire Category Theorem and the Uniform Boundedness Principle are
covered in:
A. Chapter 2 (Metric Spaces)
B. Chapter 3 (Norms and Banach Spaces)
C. Chapter 4 (Further Results on Banach Spaces)
D. Chapter 5 (Inner Products and Hilbert Spaces)
Rationale: The table of contents confirms that Chapter 4 covers "Further Results on
Banach Spaces," which includes these advanced topics.
4. Heil's textbook covers the Spectral Theorem for:
A. All bounded linear operators on Hilbert spaces
B. Compact self-adjoint operators on Hilbert spaces
C. Unitary operators on Banach spaces
D. Normal operators on finite-dimensional spaces
Rationale: The Spectral Theorem for compact self-adjoint operators is covered in
Chapter 7. The general spectral theorem for bounded normal operators requires more
advanced machinery.
5. According to Heil, the three main families of spaces encountered in analysis are:
A. Vector spaces, topological spaces, and measure spaces
B. Metric spaces, normed spaces, and inner product spaces
C. Banach spaces, Hilbert spaces, and Sobolev spaces
D. Finite-dimensional spaces, separable spaces, and reflexive spaces
Rationale: This is stated explicitly in the preface: the text is "a self-contained
introduction to these spaces" — metric, normed, and inner product spaces.
6. The dual of ℓ^p (for 1 < p < ∞) is:
,A. ℓ^q where 1/p + 1/q = 1
B. ℓ^p itself
C. ℓ^∞
D. c₀
Rationale: This is a standard result covered in Chapter 4. The dual of ℓ^p is isometrically
isomorphic to ℓ^q for 1 < p < ∞.
7. A Schauder basis for a Banach space is:
A. A Hamel basis (every vector is a finite linear combination)
B. A sequence such that every vector has a unique convergent series
representation
C. Any linearly independent set that spans the space
D. A maximal orthonormal set
Rationale: Unlike a Hamel basis, a Schauder basis allows infinite series representations.
This is a key topic in Chapter 4 that is "rarely presented in an accessible way to
undergraduate students".
8. The text requires knowledge of measure theory:
A. Throughout all chapters
B. In Chapters 1–7
C. No — measure theory is not required for the main text
D. Only in Chapter 6
Rationale: The text is designed so that "no familiarity with measure theory is required."
An optional online Chapter 8 (Integral Operators) covers related results that do depend
on Lebesgue measure.
9. Chapter 5 covers:
A. Norms and Banach Spaces
B. Metric Spaces
C. Inner Products and Hilbert Spaces
D. Operator Theory
Rationale: The table of contents confirms: Chapter 5 is "Inner Products and Hilbert
Spaces".
10. The online Extra Chapter 8 covers:
, A. Spectral Theory
B. The Uniform Boundedness Principle
C. Integral Operators
D. Fourier Series
Rationale: Chapter 8, "Integral Operators," is posted online and requires measure
theory.
11. Heil recommends a course covering all topics in:
A. One semester
B. Two semesters
C. Three quarters
D. One year
Rationale: The author recommends a 2-semester course if all topics are to be covered.
12. The notation L(X, Y) denotes:
A. The set of all linear transformations from X to Y
B. The set of all bounded linear operators from X to Y
C. The set of all continuous functions from X to Y
D. The Lebesgue space of functions from X to Y
Rationale: L(X, Y) denotes the space of bounded linear operators, which is itself a
normed space when equipped with the operator norm.
13. The dual space X of a normed space X is:*
A. The set of all linear functionals on X
B. The set of all bounded (continuous) linear functionals on X
C. The set of all bounded linear operators from X to X
D. The completion of X
Rationale: X* = L(X, F), the space of bounded linear functionals. It is always a Banach
space.
14. Which theorem states that a normed space X is a Banach space if and only if
every absolutely convergent series converges?
A. The Baire Category Theorem
B. The Uniform Boundedness Principle
