Content of the lectures and classes
Lecture 1 - 3.10.2016
Introductory information - a brief content of the course, assumed knowledge and connections to other
areas of mathematics. Beginning of Section V.1 - till Example V.1(2).
Class 1 - 3.10.2016
Introductory information on credits, Examples V.1(3)-(5),(7) (example (4) only briefly mentioned;
to finish example (7) it remains to show local non-convexity)
Lecture 2 - 5.10.2016
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)
Lecture 3 - 10.10.2016
Completion of the proof of Theorem 4(2), Section V.2 (continuous and bounded linear mappings) -
Propositions 6 and 7, definition of boundedness in a TVS, implication (i) implies (ii) from Proposition 8)
Class 2 - 10.10.2016
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; Examples V.5(1,2), within this
a charakterization of convex sets in vector spaces (A is convex if and only if
(α+β)A=αA+βA for each α,β>0), the strongest locally convex topology, all the linear
functionals are continuous in it.
Lecture 4 - 12.10.2016
Proposition 8 and a proof of the implication (ii) implies (i) for space with translation invariant metric,
definition of isomorphism into, onto, isomorphic spaces. Section V.3 - spaces with finite and infinite dimension (the whole section).
Lecture 5 - 17.10.2016
Section V.4 (metrizability of TVS) - the proof of Propostion 13 was not done, only the construction of the fuction p, properties
were not proved. Section V.5 (Minkowski functionals, seminorms and generating of locally convex topologies)
- definition, Proposition 15, Lemma 16, Proposition 17 (only the second and third assertions proved up to now).
Class 3 - 17.10.2016
The convex hull of a balanced set is balanced, the balanced hull of a convex set need not be convex;
a necessary condition for a translation invariant metric to generate a linear topology, its failure for
the discrete metric; representation of continuous linear functionals on lp
for p∈(0,1) by elements of l∞; continuous linear functionals
on the space of test functions are exactly the distribution; 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)
Lecture 6 - 19.10.2016
Completion of the proof of Proposition 17, continuation of Section V.5 till Theorem 23.
Lecture 7 - 24.10.2016
Proposition 24, then Section V.6 (F-spaces and Fréchet spaces) - till Proposition 28.
Example 25(2) was not proved, in the assertion (3) only the first case was proved, the proof
of the remaining cases is similar.
Class 4 - 24.10.2016
Translation invariant metric which generate linear topologies; preserving of boundedness and total boundedness
to the union, to the sum, to the closure and to the balanced hull; the balanced hull of a compact set
is compact; the convex hull of a compact set in a finite-dimensional TVS is compact; completeness of Lp(μ) for p∈(0,1)
(i.e., Example V.25(2)); 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 be bounded (a hint for construction an example in Lp([0,1])).
Lecture 8 - 26.10.2016
Completion of Section V.6; Section V.7 till Corollary 34.
Lecture 9 - 31.10.2016
An addendum to the proof of Theorem 31; completion of Section V.7 (Theorem 35 and Corollary 36);
beginning of Chapter VI (weak topologies), especially of Section VI.1 (general weak topologies) -
introductory definitions and Proposition 1(1-3).
Class 5 - 31.10.2016
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; 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; the metric on the space of (equivalence classes of)
measurable functions on [0,1] defined by ρ(f,g)=∫01
min{1,|f(x)-g(x)|} dx generates a linear topology; the identity mapping from C[0,1]
with the topology of pointwise convergence to C[0,1] with the metric ρ is
a bounded sequentially continuous mapping which is not continuous.
Lecture 10 - 2.11.2016
Completion of Section VI.1 (from Proposition 1(4) to the end, in Examples 2 only (1)-(3) were explained);
Section VI.2 (weak topologies on LCS) - till Theorem 8.
Lecture 11 - 7.11.2016
Completion of Section VI.2 (Proposition 9);
Section VI.3 (polars and their application) - till Theorem 15, including the first part of its proof.
Class 6 - 7.11.2016
Topology of pointwise convergence on the space of continuous functions, 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; comparison of weak and weak* topologies on the dual
to a normed space; the weak closure of the sphere in an infinite-dimensional normed space
is the ball; an example of a norm closed convex subset of l1=(c0)*, which is not
weak* closed; weak convergence of the canonical vectors in c0 and in lp
(p∈(1,∞)), weak closedness of the set of the canonical vectors in l1;
weak convergence of an orthonormal sequence in a Hilbert space; separability of a normed space is
equivalent to the weak separability
Lecture 12 - 9.11.2016
Completion of Section VI.3 (the remaining part of the proof of Theorem 15; the remaining part of the section);
beginning of Chapter VII (elements of vector integration) and Section VII.1 (measurability of vector-valued functions)
- till Proposition 1(b).
Lecture 13 - 14.11.2016
Section VII.1 from Proposition 1(c) to Theorem 5.
Class 7 - 14.11.2016
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; applicatio of this facts to a charakterization of
weak* (resp. weakly) convergent sequences as bounded pointwise convergent sequences;
a comment on the Schur theorem saying that
on l1 weak an norm convergences of sequences coincide; comparison of the weak topology
and the topology of pointwise convergence on C([0,1]) (a pointwise convergent sequence need
not be bounded; a sequence is weakly convergent if and only if it is uniformly bounded and pointwise
convergent; the weak topology and the topology of pointwise convergence differ on the unit ball);
if fn are strongly measurab weakly convergent almost everywhere to
f, then f is strongly measurabel; Example VII.6; a remark on the implication
Borel measurable implies weakly measurable and its relationship to real-valued measurable cardinals.
Lecture 14 - 16.11.2016
Section VII.2 (integrability of vector-valued functions) till Proposition 10 (the proof of Proposition 7(a) was not done as it is completely
standard).
Lecture 15 - 21.11.2016
Completion of Section VII.2 - from Proposition 11 till the end.
Section VII.3 (Lebesgue-Bochner spaces) till Theorem 14(a) (the proof of the case
p=∞ of Theorem 14(a) was not done, as it is easy).
Class 8 - 21.11.2016
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.
Measurability and integrability of functions t↦ψ(t)·χ(0,t) and t↦ψ·χ(0,t).
Lecture 16 - 23.11.2016
Completion of Section VII.3 - from Theorem 14(b) till the end.
Measurability and integrability of functions t↦ψ·χ(0,t) and t↦χ(0,ψ(t)).
Lecture 17 - 28.11.2016
Beginning of Chapter VIII (Banach algebras and Gelfand transform), Section VIII.1 (Banach algebras -
basic notions and properties) - till Proposition 4. Examples 1(8,9) were omitted.
Class 9 - 28.11.2016
Measurability and integrability of the function t↦χ(0,ψ(t)) (completion).
Examples of Banach algebras - alebras with left units, with right units;
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; the algebra with the trivial product and adding a unit to it; the algebra l1(G)
where G is a commutative group (this is a more general version of Example 1(8)).
Lecture 18 - 30.11.2016
Completion of Section VIII.1 - from the definition of an inverse element till the end of the section.
Section VIII.2 (Spectrum and its properties) - till Proposition 8(iv).
Lecture 19 - 5.12.2016
Continuing Section VIII.2 - from Proposition 8(v) to Corollary 13. The remark after Theorem 9 were only
briefly mentioned. At the end the trivial part of Proposition 14 was shown.
Class 10 - 5.12.2016
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)).
Representation of the algebra l1(Zn) using matrices, invertible elements
and spectrum, especially in
l1(Z2); spectrum and resolvent function of a nilpotent element of
an algebry, application to Jordan cells; spectrum and resolvent function of an idempotent element of an algebra.
Lecture 20 - 7.12.2016
Completion of Section VIII.2 - Proposition 14 and Corollary 15. Section VIII.3 (Holomorphic functional
calculus) - Proposition 16, definition of the holomorphic calculus and a part of the proof of Theorem 17.
Lecture 21 - 12.12.2016
Completion of Section VIII.3 - a further part of the proof of Theorem 17 (a detailed proof can be found at the lecture notes).
Section VIII.4 (Ideals, complex homomorphisms and Gelfand transform) - till Proposition 21
including the proof for unital Banach algebras. Examples 19 were only briefly mentioned.
Class 11 - 12.12.2016
Adding a unit to the algebras C0(T) and K(X);
application of the holomorphic functional calculus to a diagonal matrix, to a nilpotent elemetn,
to a Jordan cell; holomorphic calculus in the algebra C(K); complex homomorphisms on C(K), on l1(G), in particular on
l1(Z).
Lecture 22 - 14.12.2016
Completion of Section VIII.4 - from the second part of Proposition 21
to the end of the section.
Lecture 23 - 19.12.2016
Section VIII.5 (C*-algebras - basic properties) - till Proposition 30. Examples 25(2,5) were only briefly mentioned,
and only a basic scheme of the proof of Proposition 29 was done.
Class 12 - 19.12.2016
A brief information 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 l1(Z) using Gelfand transform, l1(Z)
and the Wiener algebra of the functions with absolutely convergent Fourier series. Further,
Example VIII.31 and its use to the proof of Corollary VIII.35.
Lecture 24 - 21.12.2016
Completion of Section VIII.5 - Proposition 32 and Theorem 33, brief comments on Corollary 34.
Section VIII.6 (continuous functional calculus) - Proposition 36, 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.
Lecture 25 - 4.1.2017
Section IX.1 (various types of operators on a Hilbert space) - till Proposition 5. The assertions (d),(e) from
Proposition 2 were skipped, the proof of Lemma 3 can be done by a direct computation which was not made.
Lecture 26 - 9.1.2017
Completion of Section IX.1 - from Proposition 6 till the end of the section. The `moreover part' of Theorem 8
was not proved, Proposition 10 was skipped.
Class 13 - 9.1.2017
An isometry on a real Hilbert space with trivial numerical range.
Diagonal operators on l2(Γ) - spectrum, normality, a characterization
of compactness, application of the contonious calculus; relationship to the Hilbert-Schmidt theorem;
Proposition IX.10 as a consequence.
Operators on L2(μ) defined by multplication by a bounded function - spectrum,
normality, application of the contonious calculus. Normal operators are unitarily equivalent
to multipliers (no proof given), illustration on the shift operator on l2(Z).
Lecture 27 - 11.1.2017
Section IX.2 (measurable calculus and spectral decomposition). Proposition 12 was not proved. Construction of the spectral measure,
Proposition 13 briefly commented, construction of the measurable calculus, a proof of its multiplicativity (as an illustration of the
method of the proof of Theorem 15), definition of the spectral measure of an operator and proof of properties (i) and (v) of an abstract
spectral measure, definition of the integral of a bounded function with respect to a spectral measure,
brief comments on Theorem 18 and Corollary 20.
