- Define the following:
- domain, codomain, image, range for a function
- continuity of a function at a point
- continuity of a function on its domain
- boundedness of a function on its domain
- linearity of a function on its domain
- injectivity (one-to-one property) of a function on its domain
- surjectivity (onto property) of a function on its domain
- graph of a function
- Define the following:
- linear space
- subspace of a linear space
- metric (distance)
- metric space
- norm (length) in a linear space
- a normed space
- inner (scalar) product in a linear space
- an inner-product space
- orthogonality in an inner-product space
- a sequence in a metric space
- a sequence in a normed space
- limit points of a sequence in a metric space
- convergence of a sequence in a metric space
- convergence of a sequence in a normed space
- Cauchy sequence in a metric space
- completeness of a metric space
- dense subspace of a linear space
- Define the following:
- Banach space (complete, normed linear space)
- Hilbert space (complete, inner-product space)
- triangle inequality in a normed space
- triangle inequality in a metric space
- parallelogram law in a normed space
- linear operator between two function spaces
- linear operator between two normed spaces
- bounded linear operator between two normed spaces
- operator norm of a linear operator between two normed spaces
- domain of a linear operator between two normed spaces
- range of a linear operator between two normed spaces
- continuity of a linear operator between two normed spaces
- space of bounded linear operators between two normed spaces
- unbounded linear operator
- graph of a linear operator
- densely defined linear operator
- Provide examples for the following:
- a finite-dimensional linear space
- an infinite-dimensional linear space
- a converging sequence
- a non-converging sequence
- a Cauchy sequence
- a Cauchy sequence that does not converge
- a complete metric space
- a metric space that is not complete
- a bounded linear operator between two normed spaces
- an unbounded linear operator between two normed spaces
- Elaborate on the following:
- contrast between calculus and functional analysis
- contrast between a convergent sequence and a Cauchy sequence
- contrast between a vector and a function
- contrast between a function and an operator
- contrast between a bounded function and a bounded linear operator
- contrast between a continuous function and a continuous linear operator
- contrast between a Riemann integral and a Lebesgue integral
- contrast between bounded operators and unbounded operators
- domain of a bounded linear operator vs domain of an unbounded linear operator
- Describe the following:
- L^{1}(Ω),
L^{2}(Ω),
L^{p}(Ω),
L^{∞}(Ω)
- ||f||_{1}, ||f||_{2}, ||f||_{∞}, ||f||_{p}
- C(Ω), C^{1}(Ω), C^{p}(Ω), C^{∞}(Ω)
- C_{0}(Ω), C_{c}(Ω), C^{∞}_{c}(Ω)
- ℓ^{1}, ℓ^{2},
ℓ^{p}, ℓ^{∞}
- c, c_{0}, c_{00}
- State whether true or false, make the statements precise, and illustrate with some examples:
- every convergent sequence is a Cauchy sequence
- every Cauchy sequence is convergent
- every Hibert space is a Banach space
- every Banach space is a Hilbert space
- every Banach space can be viewed as a metric space
- every metric space can be viewed as a Banach space
- every Hibert space can be viewed as a metric space
- every metric space can be viewed as a Hilbert space
- a normed space can be viewed as an inner product space if and only if
the parallelogram law holds
- every scalar product induces a norm
- every norm induces an inner product
- Elaborate on the following both in a finite dimensional vector space and
in an infinite dimensional linear space:
- advantage of an orthogonal basis over a nonorthogonal basis
- advantage of an orthonormal basis over an orthogonal basis
- can we have an orthogonal basis in a Banach space?
- can we have an orthogonal basis in a Hilbert space?
- why is relevant to know if a vector belongs to a particular vector space?
- why is relevant to know if a function belongs to a particular linear space?
- some concepts relevant to square matrices, but not relevant to nonsquare matrices
- how can we generalize the concepts of determinant and trace of a square matrix in
an infinite dimensional setting
- why are selfadjoint matrices so useful?
- what are some relevant properties of eigenvalues and eigenvectors of selfadjoint matrices?
- can we diagonalize a selfadjoint matrix via a unitary matrix?
- how can we generalize the concept of the adjoint in an infinite dimensional setting?
- why are selfadjoint operators so useful?
- what is the analog of a square matrix in an infinite dimensional setting?
- when can we diagonalize a matrix?
- what is the analog of diagonalization in an infinite dimensional setting?
- can we have a vector space where there is a vector with infinite length?
- can we have a linear space where there is a bounded function of infinite norm?
- can we have a linear space where there is a an unbounded function of infinite norm?
- can we have a linear space where there is a an unbounded function of finite norm?
- compare boundedness and finite length in a finite dimensional vector space and also
in an infinite dimensional vector space
- Elaborate on the following both in a finite dimensional vector space and
in an infinite dimensional linear space:
- Fredholm alternative
- solution to a linear homogeneous equation, solution to a linear nonhomogeneous equation
- solution to a linear homogeneous system, solution to a linear nonhomogeneous system
- Prove the following:
- 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