Aktosun, Fall 2023, Math 5327, Sample Problems 1

  1. Define the following:
    1.     domain, codomain, image, range for a function
    2.     continuity of a function at a point
    3.     continuity of a function on its domain
    4.     boundedness of a function on its domain
    5.     linearity of a function on its domain
    6.     injectivity (one-to-one property) of a function on its domain
    7.     surjectivity (onto property) of a function on its domain
    8.     graph of a function

  2. Define the following:
    1.     linear space
    2.     subspace of a linear space
    3.     metric (distance)
    4.     metric space
    5.     norm (length) in a linear space
    6.     a normed space
    7.     inner (scalar) product in a linear space
    8.     an inner-product space
    9.     orthogonality in an inner-product space
    10.     a sequence in a metric space
    11.     a sequence in a normed space
    12.     limit points of a sequence in a metric space
    13.     convergence of a sequence in a metric space
    14.     convergence of a sequence in a normed space
    15.     Cauchy sequence in a metric space
    16.     completeness of a metric space
    17.     dense subspace of a linear space

  3. Define the following:
    1.     Banach space (complete, normed linear space)
    2.     Hilbert space (complete, inner-product space)
    3.     triangle inequality in a normed space
    4.     triangle inequality in a metric space
    5.     parallelogram law in a normed space
    6.     linear operator between two function spaces
    7.     linear operator between two normed spaces
    8.     bounded linear operator between two normed spaces
    9.     operator norm of a linear operator between two normed spaces
    10.     domain of a linear operator between two normed spaces
    11.     range of a linear operator between two normed spaces
    12.     continuity of a linear operator between two normed spaces
    13.     space of bounded linear operators between two normed spaces
    14.     unbounded linear operator
    15.     graph of a linear operator
    16.     densely defined linear operator

  4. Provide examples for the following:
    1.     a finite-dimensional linear space
    2.     an infinite-dimensional linear space
    3.     a converging sequence
    4.     a non-converging sequence
    5.     a Cauchy sequence
    6.     a Cauchy sequence that does not converge
    7.     a complete metric space
    8.     a metric space that is not complete
    9.     a bounded linear operator between two normed spaces
    10.     an unbounded linear operator between two normed spaces

  5. Elaborate on the following:
    1.     contrast between calculus and functional analysis
    2.     contrast between a convergent sequence and a Cauchy sequence
    3.     contrast between a vector and a function
    4.     contrast between a function and an operator
    5.     contrast between a bounded function and a bounded linear operator
    6.     contrast between a continuous function and a continuous linear operator
    7.     contrast between a Riemann integral and a Lebesgue integral
    8.     contrast between bounded operators and unbounded operators
    9.     domain of a bounded linear operator vs domain of an unbounded linear operator

  6. Describe the following:
    1.     L1(Ω), L2(Ω), Lp(Ω), L∞(Ω)
    2.     ||f||1, ||f||2, ||f||∞, ||f||p
    3.     C(Ω), C1(Ω), Cp(Ω), C∞(Ω)
    4.     C0(Ω), Cc(Ω), C∞c(Ω)
    5.     ℓ1, ℓ2, ℓp, ℓ∞
    6.     c, c0, c00

  7. State whether true or false, make the statements precise, and illustrate with some examples:
    1.     every convergent sequence is a Cauchy sequence
    2.     every Cauchy sequence is convergent
    3.     every Hibert space is a Banach space
    4.     every Banach space is a Hilbert space
    5.     every Banach space can be viewed as a metric space
    6.     every metric space can be viewed as a Banach space
    7.     every Hibert space can be viewed as a metric space
    8.     every metric space can be viewed as a Hilbert space
    9.     a normed space can be viewed as an inner product space if and only if the parallelogram law holds
    10.     every scalar product induces a norm
    11.     every norm induces an inner product

  8. Elaborate on the following both in a finite dimensional vector space and in an infinite dimensional linear space:
    1.     advantage of an orthogonal basis over a nonorthogonal basis
    2.     advantage of an orthonormal basis over an orthogonal basis
    3.     can we have an orthogonal basis in a Banach space?
    4.     can we have an orthogonal basis in a Hilbert space?
    5.     why is relevant to know if a vector belongs to a particular vector space?
    6.     why is relevant to know if a function belongs to a particular linear space?
    7.     some concepts relevant to square matrices, but not relevant to nonsquare matrices
    8.     how can we generalize the concepts of determinant and trace of a square matrix in an infinite dimensional setting
    9.     why are selfadjoint matrices so useful?
    10.     what are some relevant properties of eigenvalues and eigenvectors of selfadjoint matrices?
    11.     can we diagonalize a selfadjoint matrix via a unitary matrix?
    12.     how can we generalize the concept of the adjoint in an infinite dimensional setting?
    13.     why are selfadjoint operators so useful?
    14.     what is the analog of a square matrix in an infinite dimensional setting?
    15.     when can we diagonalize a matrix?
    16.     what is the analog of diagonalization in an infinite dimensional setting?
    17.     can we have a vector space where there is a vector with infinite length?
    18.     can we have a linear space where there is a bounded function of infinite norm?
    19.     can we have a linear space where there is a an unbounded function of infinite norm?
    20.     can we have a linear space where there is a an unbounded function of finite norm?
    21.     compare boundedness and finite length in a finite dimensional vector space and also in an infinite dimensional vector space

  9. Elaborate on the following both in a finite dimensional vector space and in an infinite dimensional linear space:
    1.     Fredholm alternative
    2.     solution to a linear homogeneous equation, solution to a linear nonhomogeneous equation
    3.     solution to a linear homogeneous system, solution to a linear nonhomogeneous system

  10. Prove the following:
    1.     For a linear operator between two normed spaces, the following are equivalent: boundedness, continuity, continuity at a point.


    Tuncay Aktosun aktosun@uta.edu
    Last modified: April 2, 2023