Functional analysis

Below, you can find lecture notes for the course of functional analysis I gave at the University of Warsaw during the summer semester of 2018/19. For the problem sets, see Teaching → Functional analysis.

• Lecture 1
  Examples of normed space. The Riesz lemma and its consequence that only finite-dimensional normed spaces are locally compact. The equivalence of norms in finite-dimensional spaces and the fact that a finite-dimensional subspace is always closed.
• Lecture 2
  Cauchy sequences and the notion of Banach space. The most important examples of Banach spaces, in particular, we show that \(L_p(\mu)\)-spaces are complete. Bounded linear operators between normed spaces; if the domain is finite-dimensional, then every linear operator is automatically bounded. The Hahn-Banach theorem (both in real and complex case) and its consequence that the norm of any vector can be realized by a continuous linear functional.
• Lecture 3
  Characterization of continuity of a linear functional by the closedness of its kernel. \( X^\ast\) and \(\mathscr{L}(X,Y)\) as normed spaces. Isomorphisms, isometries, embeddings, boundedness below. The dual spaces of \(c_0\), \(\ell_p\) and \(L_p[0,1]\) for \(1\leq p\leq\infty\). Riesz' theorem representing the dual \(C[0,1]^\ast\) in terms of a Riemann-Stieltjes integral with respect to a function of bounded variation.
• Lecture 4
  Representation of functionals in \(C[a,b]^\ast\) as differences of two positive functionals. The notions of vector lattice, Riesz space and Banach lattice. Normalized functions of bounded variation \(\mathrm{NBV}\). Outer/inner regular Borel measures on locally compact Hausdorff spaces; the correspondence between regular measures generated by a distribution function \(f\) of bounded variation and the fact that \(f\in\mathrm{NBV}\). The Riesz-Markov-Kakutani theorem on characterization of positive functionals on \(C_c(X)\), the space of compactly supported continuous functions on a locally compact Hausdorff space \(X\).
• Lecture 5
  Finalization of the proof of the Riesz-Markov-Kakutani theorem. The space \(C_0(X)\) of continuous functions vanishing at infinity as the completion of \(C_c(X)\). A general approach to the problem of splitting a functional into its positive and negative part. The Banach lattice structure in the dual of a normed Riesz space; the Kantorovich lemma. Variation of a mesure and its \(\sigma\)-additivity. The Hahn decomposition theorem for signed measures on \(\sigma\)-algebras. Representation of the dual space \(C_0(X)^\ast\) over \(\mathbb{R}\); continuous linear functionals correspond to regular Borel \(\sigma\)-additive measures and norm of a functional to variation of a mesure.
• Lecture 6
  The space \(\mathscr{L}(X)\) of bounded linear endomorphisms of a Banach space \(X\). Invertibility of operators of the form \(I+T\) with \(\|T\|<1\). The fact that the set of all invertible operators \(\mathscr{G}(X)\) is open in \(\mathscr{L}(X)\) and that \(T\mapsto T^{-1}\) is a \(C^1\)-diffeomorphism. Basics of spectral theory: the notions of spectrum, resolvent set and resolvent of an operator; point, continuous and residual spectra. The spectrum of \(T\in\mathscr{L}(X)\) is always a compact subset of \(\mathbb{K}\) and lies inside the ball centered at the origin and of radius \(\|T\|\). If \(X\) is over \(\mathbb{C}\), then the spectrum is nonempty. The notion of compact operator and formulation of the Riesz-Schauder theorem; some preparatory lemmas: complementability of finite-(co)dimensional subspaces, the fact that \(T-\lambda I\) is a Fredholm operator whenever \(T\) is compact and \(\lambda\neq 0\).
• Lecture 7
  Continuation of the proof that \(T-\lambda I\) is a Fredholm operator whenever \(T\) is compact and \(\lambda\neq 0\); finite ascent and descent of such operators. The notion of Fredholm operator and the Fredholm alternative. Finalization of the proof of the Riesz-Schauder theorem. Criterion for compactness of integral operators on \(C[0,1]\) with the aid of the Arzela-Ascoli theorem. An application of the Riesz-Schauder theorem to an integral operator with a specific kernel \(G(x,y)\).
• Lecture 8
  Inner products and the Cauchy-Schwarz inequality. Inner product spaces and Hilbert spaces. Parallelogram law and polarization identities. Every nonempty closed convex subset of a Hilbert space contains a unique element of minimal norm. The projection theorem; orthogonal projections associated with a closed linear subspace of a Hilbert space. The Riesz representation theorem for continuous linear functionals on Hilbert spaces. Orthonormal sets and orthonormal bases; every Hilbert space has an orthonormal basis. Orthogonalization via the Gram-Schmidt procedure.
• Lecture 9
  Fourier coefficients of an element of a separable Hilbert space and the best approximation problem. The Fourier series as the expansion with respect to an orthonormal basis. Sum of squares of the first \(n\) Fourier coefficients determines the distance from the subspace spanned by the first \(n\) vector from a given orthonormal basis; Bessel's inequality. Characterization of orthonormal bases via Parseval's identity. The Riesz-Fischer theorem: \(x\mapsto\widehat{x}\) is onto \(\ell_2(A)\). Complex measures, variation and its \(\sigma\)-additivity.
• Lecture 10
  Complex-valued measures are of finite total variation. Absolute continuity and singularity of measures. The von Neumann proof of the Lebesgue-Radon-Nikodym theorem. Radon-Nikodym derivative and the polar decomposition of a complex measure, i.e. the Radon-Nikodym derivative of \(\mu\) with respect to \(|\mu|\). An application of the Radon-Nikodym theorem in the proof of the duality \(L_p(\mu)^\ast\cong L_q(\mu)\) for \(1< p<\infty\) and any positive measure \(\mu\), and \(L_1(\mu)^\ast\cong L_\infty(\mu)\) for any \(\sigma\)-finite positive measure \(\mu\).
• Lecture 11
  Finalization of the proof of the duality \(L_p(\mu)^\ast\cong L_q(\mu)\); deriving the \(\sigma\)-finite case from the finite one. An example showing that in general \(L_1(\mu)^\ast\not\cong L_\infty(\mu)\) if \(\mu\) is not \(\sigma\)-finite. The Riesz-Markov-Kakutani theorem for \(C_0(X)^\ast\) in the complex case; continuous linear functionals correspond to regular Borel complex measures. The Baire Category Theorem and the Banach-Steinhaus theorem (uniform boundedness principle).
• Lecture 12
  Some consequences of the Banach-Steinhaus theorem; weakly bounded sets and pointwise limits of sequences of operators. The Open Mapping Theorem; if a bounded linear operator on a Banach space has range of second category, then it is automatically surjective and open. Boundedness of inverse operators and equivalence of any two comparable and complete norms on a normed space. The closed graph theorem and its application: the Hellinger-Toeplitz theorem on symmetric operators on a Hilbert space. Basics of the theory of Fourier series, trigonometric polynomials and the orthonormal system \((e^{int}\colon n\in\mathbb{Z})\).
• Lecture 13
  Completeness (linear density) of the trigonometric system in \(L_2[0,2\pi]\), proved via Fejér's theorem on uniform convergence of Cesàro sums of the Fourier series of a continuous function. Dirichlet's kernel, Fejér's kernel and their properties. The norms of the function in Dirichlet's kernel tend to infinity. An application of the Banach-Steinhaus theorem to the problem of pointwise convergence of Fourier series of continuous functions. The Riemann-Lebesgue lemma and an application of the Open Mapping Theorem which shows that not every sequence in \(c_0(\mathbb{Z})\) is the Fourier coefficient sequence of some function from \(L_1(\mathbb{T})\).
• Lecture 14
  Motivation behind the definition of Fourier transform as a limit case of Fourier coefficients when period goes to infinity. Convolution of integrable functions and some basic properties, e.g. that Fourier transform changes convolution to multiplication. The Fourier transform of a derivative and the derivative of a Fourier transform, under suitable assumptions. Some heuristic approach to the problem of inverting Fourier tranform. The Fourier transform of any integrable function is continuous and vanishes at infinity. Approximate identities in \(L_1(\mathbb{R})\) and regularizations by convolution. The Fourier inversion formula and the uniqueness theorem: if \(f\in L_1(\mathbb{R})\) and \(\widehat{f}=0\), then \(f=0\) a.e.
• Lecture 15
  Observation that Fourier transform does not give any 'symmetry' between \(L_1(\mathbb{R})\) and \(C_0(\mathbb{Z})\). Definition of the Fourier transform on \(L_2(\mathbb{R})\) which can be obtained as the \(L_2\)-limit of a sequnce of 'truncated' Fourier transforms. The Plancherel theorem; Fourier transform as a unitary operator on \(L_2(\mathbb{R})\). The Fourier inversion formula is still valid provided \(\widehat{f}\in L_1(\mathbb{R})\). An example of application of the fact that Fourier transform is an isometry in the \(L_2\)-norm.