Content of the lectures and classes
Lecture 1 - 29.9.2021
Introductory information - a brief content of the course, assumed knowledge and connections to other
areas of mathematics, recommended literature etc. Brief review of the necessary knowledge from general
topology. Beginning of Section I.1 - till Example I.1(2).
Class 1 - 30.9.2021
Examples I.1(3)-(5),(7) (example (4) only briefly mentioned;
to finish example (7) it remains to show local non-convexity)
Lecture 2 - 5.10.2021
Section V.1 - from the remark after Examples 1 to Theorem 4 (including the proof of (1), the
formulation of (2), uniqueness, construction of the respective topology, a proof that it is a
topology and that the given system is a base of neighborhoods of zero).
Class 2 - 6.10.2021
The space Lp[0,1] for p∈(0,1) - the only open convex sets are the empty set
and the whole space, the unique continuous linear functional is the zero one;
the space lp for p∈(0,1)
- any nonempty open set is unbounded in the defining metric; a characterization of convex sets
in vector spaces (A is convex if and only if
(α+β)A=αA+βA for each α,β>0); Example I.5(1); Example I.5(2)
- thes space 𝒟(Ω), the norms and the metric on 𝒟K(Ω),
topology on 𝒟(Ω), the standard convergence
in 𝒟(Ω) implies the convergence in this topology.
Lecture 3 - 7.10.2021
Completion of the proof of Theorem 4(2), Section I.2 (continuous and bounded linear mappings), beginning of Section I.3
(spaces of finite and infinite dimension) - statement of Proposition 9 and the first step of its proof.
Lecture 4 - 11.10.2021
Section V.3 - the remaining part of the proof of Proposition 9, remainder of the section.
Section V.4 (metrizability of TVS), the whole section - the proof of Propostion 13 was not done, only the construction of the fuction p, properties
were not proved.
Class 3 - 13.10.2021
Continuation from the last class - convergence in the topology of 𝒟(Ω) implies the standard convergence
in 𝒟(Ω); continuous linear functionals on the space of test functions are exactly the distribution;
representation of continuous linear functionals on lp
for p∈(0,1) by elements of l∞;
the strongest locally convex topology, all the linear
functionals are continuous in it.
The convex hull of a balanced set is balanced, the balanced hull of a convex set need not be convex.
Lecture 5 - 14.10.2021
Section I.5 (Minkowski functionals, seminorms and generating of locally convex topologies)
- till Theorem 19.
Lecture 6 - 18.10.2021
Continuation of Section I.5 - from Theorem 20 till Proposition 25(b) (only the firs part).
Class 4 - 20.10.2021
Comparison of bounded and metrically bounded sets in normed linear spaces, in a TVS generated by a translation invariant metric and in the space Lp(μ) for p∈(0,1).
Space FΓ - seminorms generating the topology, metrizability is equivalent to countability of Γ,
normability is equivalent to finiteness of Γ, for a countable Γ the natural metric is complete.
Completeness of Lp(μ) for p∈(0,1)
(till constructing the limit function, the fact that it is really a limit has not been proved yet).
Lecture 7 - 21.10.2021
Proposition 24 - completing assertion (b) and assertion 3, then Section I.6 (F-spaces and Fréchet spaces) - till Theorem 31.
In Example 25 assertion (2) and the first case of assertion (3) were proved during the classes, the second case of assertion (3) was proved
and the remaining cases were only briefly commented.
Lecture 8 - 25.10.2021
Completion of Section I.6 (i.e., Theorem 32); Section I.7 (separation theorems in locally convex spaces) till Theorem 36 (the statement
and the proof of (a) for real spaces).
Class 5 - 27.10.2021
Completeness of Lp(μ) for p∈(0,1) - the remaining part. A thorough analysis of the space
⋂p∈(0,∞)Lp(R).
Lecture 9 - 1.11.2021
Completion of Section I.7 (Theorem 36(b) for real spaces, Theorem 36 for complex spaces and Corollary 37);
beginning of Chapter II (weak topologies), especially of Section II.1 (general weak topologies and duality) -
till Lemma 3 (statement and a proof of (i)⇒(ii)⇒(iii)).
Class 6 - 3.11.2021
The space C(R) is not normable, description of its dual.
For p∈(0,1) the space lp is linearly isometric to a subspace of Lp([0,1]),
which shows that the Hahn-Banach extension theorem fails in TVS;
The absolutely convex hull of a bounded set in a LCS is bounded; the convex hull
of a bounded set in a TVS need not be bounded (an example in Lp([0,1])); the convex hull of a compact
set in a TVS need not be bounded (a hint for construction an example in Lp([0,1])).
An example of two disjoint closed
convex sets in the Banach space c0 or lp for p∈[1,∞), which cannot
be separated by a nonzero continuous linear functional.
Lecture 10 - 4.11.2021
Completion of Section II.1 (implication (iii)⇒(i) from Lemma 3, Theorem 4 and Corollary 5);
Section II.2 (weak topologies on LCS) - whole section; beginning of Section II.3 (polars and their applications)
- till Proposition 11.
Lecture 11 - 8.11.2021
Continuation of Section II.3 (polars and their application) - from the remark after Proposition 11 till Corollary 17.
Class 7 - 10.11.2021
Dual to FΓ. The space C(K) - the norm topology, the weak topology,
the topology of pointwise convergence, the topology of pointwise convergence on a subset -
continuous linear functionals in these topologies. Comparison of weak* topologies on the dual
to a normed space and on the dual to its completion - on the whole space sand on the unit ball.
Weak* topology on the unit ball of l∞ coincides with the topology of pointwise
convergence; a similar thing holds for the weak topology on lp where p∈(1,∞)
and on c0 and for the weak* topology on l1; application of this facts to a characterization of
weak* (resp. weakly) convergent sequences as bounded pointwise convergent sequences; weak convergence of the canonical vectors in c0 and in lp
(p∈(1,∞));
a comment on the Schur theorem saying that on l1 weak an norm convergences of sequences coincide.
The weak closure of the sphere in an infinite-dimensional normed space is the ball.
Lecture 12 - 11.11.2021
Completion of Section II.3 (the second part of Corollary 17 and Corollary 18); a complement on lower semicontinuous and sequentially
lower semicontinuous functions and on the attainment of a minimum;
beginning of Chapter III (elements of vector integration) and Section III.1 (measurability of vector-valued functions)
- till Proposition 1.
Lecture 13 - 15.11.2021
Completion of Section III.1 - from the remark after Proposition 1 to the end of the section.
Lecture 14 - 18.11.2021
Section III.2 (integrability of vector-valued functions) till Proposition 11.
Lecture 15 - 22.11.2021
Completion of Section III.2 - from the definition of a weak integral till the end of the section (including a remark
on unconditional convergence).
Section III.3 (Lebesgue-Bochner spaces) - definitions and remarks, Theorem 14(a,b) (in (a) only the proof of the case
p=∞ has been sketched).
Class 8 - 24.11.2021
Measurability of vector-valued functions with values in lp where p∈[1,∞) or in c0, conditions
for Bochner and weak integrability, especially for c0, examples distinguishing the
types of integral for c0.
Measurability and integrability of the function t↦ψ·χ(0,t).
Lecture 16 - 25.11.2021
Completion of Section III.3 - Theorem 14 - a proof of (a) for p<∞, assertion (c) and then till the end of the section.
Lecture 17 - 29.11.2021
Beginning of Chapter IV (Banach algebras and Gelfand transform), Section IV.1 (Banach algebras -
basic notions and properties) - till Proposition 2. Example 1(9) was omitted.
Class 9 - 1.12.2021
Measurability and integrability of the functions t↦χ(0,ψ(t)) and t↦ψ(t)·χ(0,t).
Measurability of functions with values in C(K), where K is a compact metric space.
Lecture 18 - 2.12.2021
Completion of Section IV.1 - from the remarks after Proposition 2 till the end of the section.
Beginning of Section IV.2 (Spectrum and its properties) - introductory definitions and remarks.
Lecture 19 - 6.12.2021
Continuing Section IV.2 - from Remark (3) to Theorem 12 including the first part of the proof. The remark after Theorem 9 were only
briefly mentioned.
Class 10 - 8.12.2021
Examples of Banach algebras - the algebra (Cn,||·||p)
with the pointwise multiplication and its renorming;
the algebra lp(Γ) with the pointwise multiplication; various norms on the matrix
algebra; algebras with left units, with right units and their matrix representations;
the algebra with the trivial product and adding a unit to it, representation in the algebra of operators.
Spectrum in the algebra C(K) and in the algebra C0(T), adding a unit to C0(T).
Spectrum and resolvent function of a nilpotent element of an algebra, application to Jordan cells mentioned.
Lecture 20 - 9.12.2021
Completion of Section IV.2 - Theorem 12 and the rest of the proof, then continuing to the end of the section. Section IV.3
(Holomorphic functional calculus) - Proposition 16, definition of the holomorphic calculus and the subsequent remarks.
Lecture 21 - 13.12.2021
Completion of Section IV.3 - a proof of Theorem 17 (a detailed proof can be found at the lecture notes).
Class 11 - 15.12.2021
Banach algebra l1(G) where G is a commutative group (this is a more general version of Example 1(8))
Representation of the algebra l1(Zn) using matrices, invertible elements
and spectrum, especially in l1(Z2), including the holomorphic calculus.
Application of the holomorphic functional calculus, to a nilpotent element,
to a Jordan cell; holomorphic calculus in the algebra C(K);
Spectrum and resolvent function of an idempotent element of an algebra.
Lecture 22 - 16.12.2021
Section IV.4 (Ideals, complex homomorphisms and Gelfand transform) - till Proposition 22.
Example 19(3) was only briefly commented.
Lecture 23 - 20.12.2021
Completion of Section IV.4 - from Proposition 23
to the end of the section. Beginning of Section IV.5 (C*-algebras - basic properties) - introductory definitions, Examples 25(1-4), Remark (1) on the unit
in an algebra with involution.
Class 12 - 22.12.2021
Complex homomorphisms on l1(G), the dual group. Application to l1(Z) -
the dual group is T, the Gelfand transform, description of the spectrum of a general element and of a canonical vector
using the Gelfand transform. The relationship to Fourier series and to the Wiener algebra - the range of the Gelfand
tranform is formed by functions with absolutely convergent Fourier series, the Gelfand transform is one-to-one but not
surjective, hence l1(Z) is not isomorphic to a C*-algebra. Application to l1(Zm)
- the dual group is formed by
m-th roots of 1, formula for the spectrum, the algebra is isomorphic but not isometric to a C*-algebra.
Closed ideals in C(K) (Example IV.19(3)), description of maximal ideals and of Δ(C(K)),
the Gelfand transform is the identity mapping.
Lecture 24 - 3.1.2022
Continuation of Section IV.5 - Remarks (2) and (3), further from Proposition 26 to Proposition 32.
Remark: This lecture took place online via zoom, the recording is available in Moodle.
Class 13 - 5.1.2022
An example showing that the spectrum with respect to a subalgebra may be really larger (essentially Problem 25
to Chapter IV). The spectrum with respect to a C*-subalgebra does not increase (i.e., Proposition IV.36 for unital
algebras), a comparison with the preceding example.
A brief information on compact and locally compact abelian groups, on Haar measure and
on the convolution algebra L1(G) (which is a more general version of Example 1(7-9)),
on complex homomorphisms on L1(G), the dual group, applications
for G=Rn,T,Z, relationship of the Gelfand transform to the Fourier transform and Fourier
series. The spectrum of an element of a finite-dimensional commutative Banach algebra (Problem 45 to Chapter 4).
A proof of Corollary IV.35.
Lecture 25 - 6.1.2022
Completion of Section IV.5 - Theorem 33 and Corollary 34.
Section IV.6 (continuous functional calculus) - Proposition 36 (proof of (a), the part (b) has been proved during classes),
Theorem 37 stated without a proof,
construction and properties of the continuous functional calculus for unital C*-algebras, i.e., Theorem 38.
Theorem 39 was only briefly mentioned.
Application of functional calculus to matrices.
