Themes

This page lists the main definitions, propositions, examples and skills that are relevant for the course. It is intended as a check list.

Do not forget the problem sets.

• of a set; cardinality (for finite and countable sets)
• unions, intersections, (relative) complements; cartesian products
• \(\mathbb N\), \(\mathbb Z\), \(\mathbb R\), \(\mathbb C\).
• of a function (domain, codomain, graph); range of a function
• surjectivity (onto), injectivity (one-to-one) and bijectivity (invertibility)
• the inverse of an invertible function

• to express a set using curly bracket-notation (\(\{ \}\)); to find the cardinality of a set; to determine intersections, unions and complements.
• to be able to use basic quantifiers.
• to understand functional notation; differences between domain/codomain, the function itself, and its values (\(f\colon X \to Y, x \mapsto f(x)\))
• to determine the range of a function.
• to determine whether a function, or some restriction of it, can be inverted (in reasonable cases: to find its inverse).

• A bijection \(\mathbb N \to \mathbb Z\); \(\mathbb N \to \mathbb N^2\).
• The operators \(1 \pm \partial_x^2\) on \(C_{2\pi-\text{per}}^\infty(\mathbb R, \mathbb R)\).

• of a metric; metric space
• induced metric; metric subspace
• \(B_r(x_0)\), \(S_r(x_0)\); the closed ball \(\tilde B_r(x_0)\)
• interior points, boundary points, closures; open and closed sets
• limits (continuous, sequential); accumulation points
• Cauchy sequences; completeness
• Isometries; embeddings
• Dense sets and separability
• Continuous mappings; contractions.

• Normed spaces are metric spaces
• (Open) balls are open
• Uniqueness of limits; Sequential limits are as zero-limits for the distance function; Continuous and sequential limits agree
• Characterization of closures in terms of limits
• Convergent sequences are Cauchy; Cauchy sequences are bounded
• Characterization of complete subspaces of complete metric spaces
• The completion theorem

• to understand the difference between a norm and a metric; between a normed space and a metric space.
• to determine whether a function defines a distance on some set \(X\).
• to determine whether a sequence is convergent; whether a sequence is Cauchy.
• to find the boundary of a set; to determine whether a set is open/closed.
• to know the main steps in the completeness proof for \(BC(I,\mathbb R)\).

• The discrete metric
• A metric space that is not a normed space
• To know that \(BC(I,\mathbb K)\) and \(l_p\) are complete.
• A non-trivial isometry
• To know that polynomials and \(C^k\)-functions are dense in \(BC\)-spaces (Stone–Weierstrass).
• A non-separable and some separable spaces.
• Completion of \(\mathbb Q\) in \(\mathbb R\); of \(C^\infty\)-functions in \(BC\)-spaces; of continuous functions in \(L_2\)-spaces.

• of a real (complex) vector space, subspace
• isomorphisms, embeddings
• linear combinations, spans, dependence, independence
• (Ordered) Hamel bases; dimension of a vector space
• Change-of-basis matrices
• Linear systems, (reduced) row echelon form, upper/lower triangular matrices, diagonal matrices; augmented matrix
• Linear transformations; the vector spaces \(L(X,Y)\) and \(L(X)\)
• Kernels and ranks; the null space, columns space and row space of a matrix
• Direct sums
• Nilpotent operators/matrices
• Convex sets

• Linear span of a set is a subspace (smallest one containing the set).
• Finite-dimensional vector spaces are isomorphic to \(\mathbb R^n\) (\(\mathbb C^n\) if complex).
• Invertible matrices correspond to bases.
• \(L(X,Y)\) is a vector space.
• Linear transformations are determined by their action on any basis; finite-dimensional linear transformations correspond to matrices.
• The kernel and range of a linear transformation are vector spaces; a linear transformation is injective exactly if it has a trivial kernel.
• The rank–nullity theorem (algebraic and geometric version)
• Characterization of invertible linear transformations on a finite-dimensional vector space
• The Fredholm alternative

• to understand the difference between a real and complex vector space.
• to know how to construct a vector space; determine whether a set is a subspace
• to determine linear dependence/independence
• to be able to switch from one basis to another (expressing vectors and linear transformations/matrices in the new basis).
• Gaussian elimination, LU-decompositions, Gauss-Jordan elimination.
• To calculate the determinant of a matrix.

• \(\mathbb R^n, \mathbb C^n\)
• \(C(I,\mathbb R), C(I,\mathbb C)\)
• \(P_n(\mathbb R)\), \(P(\mathbb R)\)
• \(P_n(\mathbb R) \cong \mathbb{R}^{n+1}\)
• \(P_n(\mathbb R) \hookrightarrow \mathbb{R}^{m}\) for \(m \geq n+1\); \(C^{m}(I,\mathbb R) \hookrightarrow C^n(I,\mathbb R)\) for \(m \geq n\).
• Linearly independent sets in \(\mathbb R^n\), \(\mathbb C^n\), \(l_2\) (real or complex), \(L_2((-\pi,\pi),\mathbb K)\) (real and complex), and \(P(\mathbb R)\).
• Canonical (standard) bases
• Dimensions of standard spaces
• The kernel and range of \(d/dx\) as an operator between different vector spaces.

• of a norm; normed space
• equivalence of norms
• of a Banach space
• isometrical isomorphisms
• (Ordered) Schauder bases
• Bounded linear transformations, the operator norm, and the space \(B(X,Y)\).
• Bounded linear functionals and the dual of a normed space.

• Any two norms on \(\mathbb R^n\) are equivalent.
• Equivalent expressions for the operator norm.
• \(B(X,Y)\) is a normed space, Banach for \(Y\) Banach.
• A linear operator between normed spaces is continuous if and only if it is bounded.
• The kernel of a bounded linear operator is closed.

• To determine whether a function is a norm.
• To use the axioms of a norm.
• To determine whether a sequence (vector) is in \(l_p\); a function is in \(BC(I,\mathbb R)\).
• To determine whether two norms are equivalent.

• \(l_p\)-spaces (including their norms; especially \(p=1,2, \infty\))
• \(BC(I,\mathbb R)\) (including the definition of the supremum norm \(\|\cdot\|_\infty)\)
• Standard Schauder bases for \(l_p\), and \(L_2\) on the interval \((-\pi,\pi)\)
• Integral operators as bounded linear transformations
• The dual of \(\mathbb R^n\); of \(L_2(I,\mathbb R)\).
• Finite-dimensional linear transformations are continuous.

• Initial-value problems
• Lipschitz continuity (at least uniform Lipschitz continuity)
• Contractions
• Eigenvalues, eigenvectors and eigenspaces; spectrum and resolvent set.
• Characteristic polynomials; algebraic and geometric multiplicities; simplicity and semi-simplicity of eigenvalues.
• Fundamental systems/matrices.
• The matrix exponential; nilpotent matrices.
• Generalized eigenvectors and generalized eigenspaces; the maximal generalized eigenspace; the Riesz index.
• The transpose and conjugate transpose of a matrix; Hermitian (self-adjoint) matrices.

• Higher-order ordinary differential equations can be reformulated as first-order systems.
• Lipschitz continuity is uniform on bounded and closed sets in \(\mathbb R^n\).
• The Banach fixed-point theorem (Picard iteration)
• The Picard–Lindelöf theorem
• The solution space of \(\dot x = Ax\) in \(\mathbb K^n\) is isomorphic to \(\mathbb K^n\).
• Properties of the matrix exponential
• The exponential solution formula for \(\dot x = Ax\), \(x(0) = x_0\).
• The Cayley–Hamilton theorem
• Each eigenvalue has a maximal generalized eigenspace.

• To transform an ordinary differential equation into a first-order system
• To apply the Banach fixed-point theorem to solve fixed-point problems (such as the initial-value problem); to perform Picard iteration.
• To calculate the matrix exponential for a given matrix.
• To express a matrix in Jordan normal form.
• To solve the initial-value problem \(\dot x = A x\), \(x(t_0) =x_0\) in \(\mathbb R^n\).

• Examples showing that \(C^1(\mathbb R) \subsetneq Lip(\mathbb R) \subsetneq C^0(\mathbb R)\).
• On the spectral and Jordan decompositions.

• Inner products; inner-product spaces
• Hilbert spaces
• Orthogonality and orthogonal complements
• Orthogonal direct sums (\(\oplus\)) in Hilbert spaces
• For sequences and finite sets (systems): orthogonal, orthonormal, complete; Fourier coefficients and Fourier series
• Orthonormal bases
• Adjoints (transposes and conjugate transposes for matrices); self-adjoint operators (symmetric and Hermitian matrices).
• Unitary operators (unitary and orthogonal matrices)
• Positive definite and semi-definite matrices

• The Riesz representation theorem
• Properties of an inner product (apart from the axiomatic properties)
• The Cauchy–Schwarz inequality
• Inner-product spaces are normed spaces.
• Relations between inner products and norms (the parallelogram law and polarization identity).
• The minimal distance theorem
• The projection theorem (corollary: strict subspace characterization)
• Fourier coefficients are best possible (corollary: closest point)
• The Fourier series theorem
• Properties of adjoints
• Self-adjoint operators have real spectrum and orthogonal eigenspaces.
• Unitary operators preserve inner products.
• Orthogonal matrices describe ortonormal bases.
• Characterization of positive definite matrices

• To determine inner products, and whether a norm comes from an inner product
• To apply the Cauchy–Schwarz inequality
• To determine minimal distances and minimizing elements
• To understand and determine orthogonality in different inner-product spaces
• To apply the projection theorem and its corollary
• To determine/construct finite orthonormal sets.
• Gram–Schmidt orthogonalization and QR-decomposition

• Standard inner products on \(\mathbb R^n\) and \(\mathbb C^n\)
• Pythagoras theorem
• The Hilbert spaces \(l_2(\mathbb K)\) and \(L_2(I,\mathbb K)\).
• Pre-Hilbert spaces constructed using the \(L_2\)-norm (there are different ones, depending on which functions one starts with).
• Linear subspaces are convex sets.
• Minimization of distances in \(L_2((-\pi,\pi),\mathbb C)\) using the standard Fourier basis \(\{e^{ikx}\}_{k\in \mathbb Z}\)
• The Rank–nullity theorem in light of the projection theorem
• \(\overline{l_0} = l_2\) in \(l_2\).
• The duals of \(\mathbb R^n\), \(\mathbb C^n\) and \(L_2(I,\mathbb R)\)
• Orthonormal sequences and bases in standard Hilbert spaces
• Finding closest points using Fourier coefficients