# Content of the lectures

### Oct. 19

Introduction. Importance of functional analysis. Informal discussion of the five basic (linear) partial differential equations and how to relate them to standard linear algebra. Think of functions as points (vectors) of a big linear space. Intuitive derivation of the linear transport, Laplace, heat and wave equations. Schrodinger equation mentioned. If you want to read a little bit more, the material I presented was taken from Chapter 2 of Evans: Partial Differential Equations (AMS 1998: Graduate Studies in Mathematics), available at the library. This material is only for broadening your horizon, starting from next class we start theorem/proof...

### Oct. 22

Possible definitions of the exponential of a matrix (power series, diagonalization). How to generalize it to the Laplace operator to define the solution of the heat equation? The standard proof via power series requires finite norm. I introduced the supremum norm on the space of functions and showed that the Laplacian as an operator on this space has no finite norm, therefore the power series definition of the exponential function does not work. The diagonalization has a better chance. I recalled the spectral theorem for matrices.

We reviewed linear algebra. Concept of a general vectorspace, subspace, factorspace, linear map, isomorphism theorem. Linear span, independence, basis. Difference between finite and infinite dimension. Theorem: if the space has a finite spanning set, then (i) the cardinality of any linearly indep. set is smaller or equal than that of any spanning set; (ii) the space has a basis and every basis has the same cardinality. This is the easy case, it is the realm of the usual (finite dimensional) linear algebra.

Construction of the basis (two ways) in finite dimension. Emphasize the importance of the finite sums in the definition of the span and linear independence.

Definition of l^\infty, the vectorspace of bounded sequences. The unit vectors (e_i = (0,0, ... 0, 1, 0, 0, ...)) do not form a basis.

Theorem: l^\infty has a basis, but it is uncountable and nonconstructive. (Partial proof comes).

The way out is that we will have to allow infinite linear combinations, but that will require analysis (concept of convergence) otherwise we do not know what an infinite sum is. This is the starting point of functional analysis.

Hopefully all notions of finite dimensional linear algebra are well known, perhaps sometimes I used a different notation, but the concepts should be clear. If not, then you can/should tell it to Dr. Sorensen in the Exercise session, and/or come to our office hours, and/or check out my linear algebra lecture notes. This course was given to students who already had a one semester linear algebra, but they had the tendency to forget everything, so I reviewed it for them in these notes.

### Oct. 26

l^infty over the finite field of 2 elements cannot have a countable basis.

Partially ordered set, Zorn lemma, Axiom of choice. Every vectorspace has a basis (Hamel basis or algebraic basis)

Characterization of additive functions over reals.

End of the review of linear algebra: Linear maps and their representation with a matrix. Change of basis. Dimension formula.

Left and right inverse, equivalent characterizations of invertibility of a square matrix. Inverse of a product.

Left and right shifts in l^infty.

Scalar product, orthogonality, orthonormal basis (ONB) and its advantages. Orthogonal and unitary matrix (transformation)

Diagonalization, spectral theorem.

Material: Reed-Simon, Sec I.1 and Note on the existence of a basis.

### Oct. 29

Degenerate maps, pseudoinverse, index of linear maps. Product rule, stability of index. I discussed the material of the lecture notes (apart from the proof of the last theorem). Don't get scared of this, from next time we start something more down to earth, but I thought once you should see the power of linear algebra beyond the usual finite dimensional matrices.

### Nov 2,

Proof of the stability of index from lecture notes.

Then I started analysis, with a grand picture that outlined the relation between topological, metric, normed, Banach and Hilbert spaces, I started with the metric spaces. (Topological spaces in general will not be discussed). Convergence, continuity, Cauchy sequence, completeness.

The most important example, the two different metrics on C[0,1] (supremum norm and integral norm). This material is mostly found in Reed-Simon, Sec I.1, I.2

It is expected that everybody is familiar with standard epsilon/delta arguments. This is absolutely basic. If you have the slightest doubt about your ability of being able to prove, say, that the sum of two continuous functions is continuous, then seek help immediately, either with Dr. Sorensen or with me. You will suffer later tremendously if you don't practice these concepts.

Then I introduced the concept of topological spaces and a few basic definitions. You can find them in a short note.

### Nov 5,

Every metric space has a completion. The proof gives a construction that is useless in practice. One has to identify the completed space.

Two types of convergences of sequences of functions between metric spaces: pointwise and uniform. Uniform limit of cont. fn's is cont. Proof is extremely important! C[0,1] is complete with the supremum metric.

Normed space, equivalent norms. In finite dimensions all norms are equivalent. In finite dimensions the unit ball is compact. In infinite dimensions the unit ball is not compact. The proof uses two lemmas: 1, for any closed subspace U of a normed space, there exist a unit vector that is 1-delta far from U. 2, any finite dimensional subspace is closed.

Bounded linear map between normed spaces. For a linear map continuity and boundedness are equivalent.

This material is found in Reed-Simon I.2 and I.5 and in Werner: I.2.1 -- I.2.8. Some statements are known from Analysis (maybe you stated them for Euclidean spaces and not for metric spaces, but the proofs go through directly -- Check)

### Nov 9,

Definition of the Banach space

Bounded extension principle. Definition of the Riemann integral on piecewise continuous functions using the bounded extension principle. (Reed-Simon I.2)

Crash review of Lebesgue integral: We want to interchange integration and limit -- does not work for Riemann. Riemann subdivides the x-axis but then the height of the function on a small interval is not well behaved. Lebesgue, instead, subdivides the y-axis and looks at the size of the preimage set. Unfortunately there is no natural way to define the size of every subset of the Euclidean space. Banach-Tarski paradox (one can cut an apple into finitely many pieces, then rearrange these pieces by Euclidean motions into an apple of twice bigger size).

Monotone and dominated convergence theorems and Fatou lemma (no proofs). Proper definition of L^p: equivalence classes. Concept of almost everywhere. Spaces of sequences: consider the measure space L^p on the natural numbers with the counting measure.

We assume that everybody is familiar with the following concepts from Lebesgue integration theory. If not, please look them up in your Analysis III notes or in the literature given below or come to office hours.

Sigma-algebra, Borel sets of the Euclidean spaces. Borel sets are generated by open rectangles. Not every subset is Borel.

Measure. Construction of Lebesgue measure. One possible path is via outer measure and Caratheodory. It is not an easy proof to show that the sets satisfying the Caratheodory criterion form a sigma-algebra and that the outer measure restricted to this sigma algebra is a measure. See Lieb-Loss: Analysis (Thm 1.15). Check that all open rectangles in the Euclidean space satisfy the Caratheodory criterion. This leads to the Lebesgue measure.

Lebesgue measure is regular (can be approximated with open sets from outside and compact sets from inside) and is invariant under Euclidean motion. The Lebesgue measure is the unique translation invariant measure on the Borel sets (if the measure of the unit cube is normalized to one).

Measurable function, this property is closed under any imaginable operation (arithmetic, composition, limit etc.) Negligible sets (covered by a zero measure Borel set). Lebesgue sets (Borel sets plus negligible sets).

Lebesgue integral. Two definitions: (i) limit procedure (given e.g. in Reed-Simon or Rudin); (ii) Riemann integral of the level sets (given in Lieb-Loss). Basic properties of the integral. Product measure, Fubini theorem.

This material is found in Reed-Simon I.2, I.3 and in Lieb-Loss, Sec 1.2, 1.5--1.8, 1.10--1.12, 1.15 and 2.1.

### Nov 12,

Riesz-Fischer Theorem (L^p is complete).

Jensen, Holder, Young inequalities. Jensen and special case of Young (q=1) was proven. Convolution.

We have seen that L^1[0,1] is a complete normed space that contains C[0,1] with the integral norm. In order to finish the proof that the completion of C[0,1] is indeed L^1[0,1], we finally showed that C[0,1] is dense in L^p for any finite p. To prove this fact, we first showed that the simple functions are dense in L^1, then that the really simple functions are dense in the simple functions and finally, by introducing a smooth cutoff function, we showed that smooth functions can arbitrarily approximate really simple functions. The proof works not just for [0,1] but for any subset of R^d.

As a bonus, we also obtained that C_0^infty(U) (smooth, compactly supported functions) are also dense in L^1(U) where U is an open subset in the d-dimensional Euclidean space

This material is found in Reed-Simon I.3 Lieb-Loss: 1.17, 1.19, 2.2-2.3, 2.15 and 4.2

The theorem in 1.18 (approximation by really simple functions) was done in a simpler setup (not for general space, but for subsets of Euclidean spaces with Lebesgue measure, and then one could approximate any open set by a union of rectangles chosen from a sequence of grids)

### Nov 16,

Second proof of the density of the smooth compactly supported functions in L^1. We proved that f_eps:=f*j_eps converges to f in L^p whenever f is in L^p and j_eps(x)= eps^{-d} j(x/eps) with integral j(x) dx =1, and j is in L^1. The function f_eps is smooth only if j is compactly supported (I forgot to mention this extra condition!).

This is a long, but very important proof because it is an example of a very common argument in analysis: if we want to prove something for all L^p functions, in many cases it is sufficient to prove it for smooth, compactly supported functions (for which the proof is much easier), then use what is called "standard approximation argument". Usually such an argument is indeed standard, but not at all trivial. Such proof involves lots of tinkering: one usually has to cutoff both the domain and the range of the function (reduce the problem to compactly support and bounded function), then one needs smoothing (usually a convolution with j_eps)

There is always a good inequality working in the background that enables us to make estimates uniform in one limiting parameter, while taking the limit in the other. In this proof the Young's inequality was such an a-priori bound.

Non-Lebesgue measures on R. Lebesgue-Stieltjes measure, Dirac delta measure. Dirac-delta "function" as viewed by physicists. Cantor set, Cantor staircase: a measure supported on a Lebesgue-zero measure set. Fundamental theorem of calculus does not hold for the Cantor staircase function.

The approximation proof can be found in Lieb-Loss 2.15-2.16. The rest is from Reed-Simon I.4.

### Nov 19,

Pure point measure. Absolute and singular continuous measures. Radon-Nykodim theorem (every ac. measure is an integral of an L^1 function), Lebesgue decomposition theorem.

Inner product space (watch out: Reed-Simon takes conjugate linear in the first variable, Werner in the second. We'll follow RS). Primary example: L^2 of any measure space (e.g. the square integrable sequences, l^2).

Schwarz inequality. Norm. Parallelogram identity. Hilbert space. L^2 is Hilbert (since complete) by Riesz-Fischer.

Orthogonal complement. Existence of the closest point (Riesz Lemma). We did it in general setup: for convex, closed sets, in particular for closed subspaces. Converse of Riesz Lemma is an unsolved question.

The material was taken from Reed-Simon I.4, II.1-2.

### Nov 23,

Riesz Projection theorem (decomposition of Hilbert space into the direct sum of a subspace and its orthocomplement.)

Space of bounded linear maps is a Banach space if the target space is Banach. Dual space of a Hilbert space. The dual of a H-space is isometrically isomorphic to the H-space.

Absolute convergence in a Banach space implies convergence. Similar statement in Hilbert space. The limit is independent of the order. Orthonormal systems (ONS) are linearly independent (finite lin. combination is zero if all coeff. are zero). Orthonormal basis: maximal ONS.

The material was taken from R-S II.2--3. Here is a short notes on the H-space basis

### Nov 26,

An ONS is an ONB iff the set of its finite lin. comb. is dense.

Every vector in a Hilbert space is the limit of finite lin. comb. of vectors from ONB (with the obvious coeff.'s), and the square of the norm is the limit of the sum of the squares of these coeff.'s

Conversely, given a square summable sequence, we get an element in H by using this sequence as coeff's. The square of the norm is the sum of the squares.

The representation of any vector as such a limit is unique.

Fourier transform of periodic functions (functions on the torus).

Dirichlet and Fejer kernels. Fourier series of continuous periodic functions is Cesaro summable and the convergence of the Cesaro sums is uniform.

The material was taken from R-S II.1 & 3, and from the notes on the H-space basis and from the beginning of the Notes on Fourier transform

### Nov 30,

Proof that Fourier series of continuous periodic functions is Cesaro summable and the convergence of the Cesaro sums is uniform.

Thm: Fourier series of L^2 function f converge to the function f (L^2-convergence); the set {e_n} is an ONB.

Discussion of Fourier-transform; unitary equivalence between L^2 and l^2(Z).

Discussion of pointwise convergence; there exists continuous functions whos F-series does not convergence pointwise (existence-proof: later); for general L^2-function, the F-series does converge pointwise _almost everywhere_ (hard to prove!)

Thm: If f is C^1, then the F-series of f converges uniformly to f; also, the sum of n^2|c_n|^2 is finite.

If the sum of n^2|c_n|^2 is finite, it does not follow that f is C^1; but: Thm: If the sum of |c_n|n is finite, then f is C^1. - More general, if the sum of |c_n|n^k is finite, then f is C^k.

So, the F-transform does not nicely characterize differentiable functions.

Discussions of derivatives: the F-transform (which is unitary) diagonalizes derivatives: De_n=(in)e_n (here D=d/dx) (e_n and (in) are eigenvectors and -values for D); the operator -iD is better, because it is symmetric wrt. scalar-product in L^2 (D is not). Also, the Laplacian -Delta is multiplication by n^2.

The material is sect. 1.2 to 1.4 (incl.) in Notes on Fourier transform

### Dec 3,

Fourier transform diagonalizes the derivative. Sobolev spaces. Equivalent norms, fractional powers, H^{1/2+eps} is continuous. Fourier transform in several variables.

General function of the Laplace operator. Solution to the heat equation on the circle. The material is sect. 1.5-1.6 in Notes on Fourier transform

### Dec 7,

Correction to the fractional derivative (I added some remarks in the Notes on Fourier transform around page 11)

F-transform on L^1(R^n). Convolution , translation, scaling. F-transform of Gaussians. Extending F-transform to $L^2$, Plancherel identity (F. T is unitary). [The missing 2pi in the lecture is fixed in the notes]

Fourier inversion formula. Dirac delta notation.

See Notes on Fourier transform until the end. Some material was taken from Lieb-Loss, Sect 5.1--5.4 (note the different 2pi convention)

### Dec 10,

Every separable H-space is isomorphic to C^n or l^2.

Direct sum of Hilbert spaces. (example 5 of Section II.1. of Reed-Simon)

Tensor product of Hilbert spaces. (Reed-Simon Sec II.4)

Basis in the tensor product.

### Dec 14,

Tensor product of L^2 spaces. (Reed Simon, Sec II.4)

Banach spaces. Examples. Equivalent norms. Isomorphism and isometrical isomorphism of normed spaces. (Reed-Simon III 1.)

On finite dimensional normed spaces any two norms are equivalent. Counterexample in infinite dimension. (Section I.2 in Werner)

Dual of Banach spaces. (Reed-Simon III.2) Hilbert spaces are self-dual (Reed-Simon II.4) Bra-ket notation from physics.

Dual of L^p(M, mu) is L^q for sigma finite measure, if p is finite (Reed-Simon, Thm. S.4 (appendix) or Wegner: II.2.4) We proved it for l^p the general case follows. The p=1 case requires a little bit separate argument.

### Dec 17,

Completion of the proof of the dual of L^p

L^1 is included in the dual of L^\infty, but is in general smaller (proof after Hahn-Banach). Nevertheless, l^1 is the dual of c_0 (Reed-Simon, Sec III.2 Ex 3). L^1(R) is not the dual of anything (without proof).

Signed and complex measures. Definition of total variation. The total variation is a measure (without proof, see, e.g.: Rudin, Real and Complex Analysis, Thm 6.2). Positive and negative part of a measure.

Radon-Nikodym for nonnegative measures. (Reed-Simon; Theorem S1. in Supplement). Proof for finite measure case. (to be finished next time)

### Dec 21,

End of the proof of Radon-Nikodym. Generalization to complex measures and to sigma-finite measures (see: Exercise session)

Let X Banach. Is its dual, X^*, nontrivial? Answer: Hahn-Banach theorem. Definition of sublinear function, statement of Hahn Banach for real vectorspaces. (Reed-Simon, Sec III.3)

Proof with the greedy extension idea may not work. Problems: 1, transfinite induction; 2, ensuring continuity.

Difference between X' (algebraic dual) and X* (normed space dual); notation may change from book to book.

Proof of real Hahn-Banach. Zorn lemma.

Proof of complex Hahn-Banach.

Corollary: Linear functional from a subspace can be extended to the whole space without increasing the norm

Merry Christmas and don't forget the Christmas Problems (not part of the course, just for fun)

### Jan 11,

Further corollaries of Hahn-Banach: X* separates X.

Weak convergence. Limit is unique. Examples

Mazur Thm: convex closed sets are weakly closed (without proof, see Thm. III.3.8 of Werner. Special case for L^p functions Thm 2.13 of Lieb-Loss)

Separation of convex sets (Thm III.2.4 of Werner) Minkowski functional. Absorbing property. Convexity implies sublinearity of the M. functional (Lemma III.2.2 of Werner).

Material: Werner III.2 Christmas Problems are extended for next week. (not part of the course, just for fun, but it's worth playing with them)

### Jan 14,

End of the proof of the separation theorem. (proof of the lemma about separating a convex set from an outside point).

l^infty is not separable.

l^l is not the dual of l^\infty: proof that natural map does not work (Werner Satz III.1.11) - in general: l^1 separable, l^\infty not - use:

Thm: X* separable implies X separable (R-S Thm III.7). Note: converse is NOT true (ex. l^1 and l^\infty).

Def. of double dual (bi-dual) X**. Thm: X is naturally isometrically embedded (map J) in X**. Corollary: Any normed space (not necessarily complete) is isom. isom. to a dense subspace in some Banach space, namely of the closure of J(X) (natural way of completing a space). Def: If X is isom. isom to X** (VIA map J), X is called reflexive.

X is reflexive if and only if X* is reflexive (without proof) (Material: Werner Kap. III.3, R-S III.2)

Definition: If the closure of S has empty interior, then S is said to be 'nowhere dense'.

Thm (Baire category thm). No complete metric space is a countable union of nowhere dense sets. (Material: RS III.5)

### Jan 18,

Thm (Principle of Uniform Boundedness/Banach-Steinhaus): Let F be a family of bounded operators from X (Banach-space) to Y (normed space), such that for all x the set { ||Tx|| | T in F } is bounded (constant possibly depending on x). Then the set { ||T|| } is bounded. (proof)

Thm (open mapping theorem) Let T be a SURJECTIVE (onto) bounded linear operator from X to Y (both B-spaces). If M is open in X, then T(M) is open in Y.

Corollary of B-Steinhaus: Let T_n: X -> Y cont, and Tx := lim T_n x exists for all x, then T is bounded. (note that in general pointwise limit of continuous functions is not continuous)

Corollary of B-Steinhaus: Coordinatewise continuous bilinear maps are continuous.

Def: open mapping. Equivalent characterizations (Lemma IV.3.2 Werner)

Example for a non open mapping

First part of the proof of the open mapping theorem (second part next time)

The final exam is set for Feb 5 (Saturday) at 15:30. Room will be announced.

### Jan 21,

End of the proof of the open mapping theorem.

Inverse mapping theorem (Kor. IV.2.4 Werner or Thm III.11 in RS)

Injective mapping theorem (Kor.IV.2.6 Werner): an injective map between B-spaces has continuous inverse from the range iff the range is closed. Graph of a linear map. Closed map (NOT the analogue of open map, it does NOT mean that it maps closed sets to closed sets. A map is called closed if its graph is closed.) Main example: differentiation. Relation between continuity and closedness.

Closed graph theorem. Hellinger-Toeplitz theorem. (Reed-Simon III.5 and Werner IV.4)

Application: impossibility to extend differentiation onto continuous functions or onto L^2 functions in a meaningful way.

### Jan 25,

Introduction into quantum mechanics. Wave function. Difference between classical and quantum mechanics. Schrodinger equation. Impossibility to define the solution of the free Schrodinger equation via the Taylor expansion of the exponential of the Laplacian; the expansion will not define a map from L^2 to L^2. See my Crash course on quantum mechanics .

Weak convergence. Examples. For the weak convergence of a bounded sequence it is enough to check the limit for a dense subset of bounded functionals.

Application of the principle of uniform boundedness for functionals. A subset of a normed space is bounded if and only if its image under any bounded linear functional is bounded. Weakly convergent sequences are bounded (Korollar IV.2.2-2.4 from Werner). Norm of a weak limit can drop but cannot increase.

Recall Mazur theorem (without proof; Hilbert space version is proved in Exercise class) (Thm III.3.8 from Werner).

Compactness in general, vs. sequential compactness. For metric spaces they are equivalent (Satz B.1.7 Werner) but not in a general topological space. The difference between these two concepts is seen only in topological spaces which have uncountable bases of neighborhood. If you are interested, read Reed-Simon, Sec IV.3. and especially Thm IV.3, or Werner Thm. B.2.8-9.

Basic facts about compact sets (without proof): closed subset of a compact set is compact, continuous image of a compact set is compact, direct product of compact sets is compact. All these are found in any analysis book, or in Sec IV.3 of RS. The last statement is Tychonoff theorem. For countable direct product a simple Cantor diagonalization argument is sufficient (exercise!) The general case uses the axiom of choice (Werner B.2.10 or Reed-Simon IV.5., but for both you have to understand the concept of nets, which are generalizations of sequences (Reed-Simon Sec IV.2 or Werner page 476, 480). This is not required for the course, but you can read it for fun.

In metric space compact sets are closed and bounded. Converse is wrong in general (unit ball of an infinite dimensional normed space), but true in R^n (Heine-Borel).

You should also read a Extended Crash course on topology . (until Tychonoff theorem).

See the information on the FINAL EXAM on the main webpage

### Jan 28.

Def. of weak sequential compactness (w.s.c) in a Banach space.

A w.s.c set is bounded.

In a reflexive B-space the closed unit ball is w.s.c. (no proof in class, you can look it up in Werner III.3.7). Consequence of this and Mazur: In a reflexive Banach space every bounded, closed, convex set is weakly seq. compact. This is as far as Heine-Borel could be extended to infinite dimensional spaces.

All these concepts are discussed in the special case of L^p spaces in Lieb-Loss (from Def 2.9 up to the end of Thm.2.13)

Definition of w* convergence. (in a space U that is the dual of a normed space X, U=X*. X does not have to be Banach) W*-conv is Weaker than the weak convergence.

Riesz-Markov theorem: Dual of C([0,1]) is M([0,1]), ie. the space of measures with the total variation norm I did not present the proof, see Thm S.5 in the supplement of Reed-Simon for the case of the interval, general case: Thm II.2.5 Werner.

Example when weak* conv. is really weaker than weak conv (consider a sequence of approximate deltafunctions that converge to the deltafunction in the weak* sense in the Banach space M([0,1])= C([0,1])*, but the convergence does not hold weakly, because it does not hold tested against an appropriate L^infty function and L^infty is a subset of M([0,1])^*)

Weak* convergent sequence is bounded and the norm of the limit is bounded by the liminf of the norms of the elements.

Definition of weak* sequentially compact.

Helly's Theorem: Let U= X*, where X is a separable Banach space. The closed unit ball of U is w* seq. compact. Proof: Cantor diagonalization argument (see Lieb-Loss; page 68). The proof of Thm 2.18 of Lieb-Loss seems to be a special case of Helly's theorem, but it is the same proof as I presented.

General statement of Banach-Alaoglu (no proof): The unit ball of the dual of a Banach space is compact in the w* topology. (Note that Helly's Thm is a special case).

In order to make sense of compactness in w* topology, we need a Extended crash course on topology .

w* topology as the weakest topology on U=X* that makes all functions l mapped to l(x) continuous (for any x).

See the MODIFIED information on the FINAL EXAM on the main webpage

### Feb 1.

Solving a linear elliptic PDE, - D^2 u + u = f, f is given, smooth, cpct support with the help of F.transform (here D^2 is the Laplacian). Solution is regular. Mentioning existence, uniqueness and regularity questions in general.

We considered the nonlinear elliptic equation, - D^2 u + u|u| + u = f. Definition of the weak solution in L^2. Theorem: this equation has a weak solution. Mentioning that a separate theory (regularity theory of elliptic PDE) guarantees that the weak solution is actually smooth, so it is solution in the classical sense.

Minimizing functional. Minimizer satisfies the differential equation in a weak sense.

Existence of minimizer goes back to a standard theorem that a continuous function on a compact set attains its minimum.

In our case: weak convergence is the right concept, since the natural space (ball in L^2) is compact in this topology (one version of Banach-Alaouglu, in this particular case Helly thm is enough). But the functional is not continuous in the weak top.

Definition of lower semicontinuity, its equivalence to the closedness of the level sets (x: F(x) =< a). Definition of weak lower semicontinuity.

To find minimizer on a compact set, the continuity can be relaxed to lower semicontinuity.

Claim: the functional associated to the equation above is weakly lower semicont. Check term by term:

Convex, norm lower semicontinuous functionals are weakly lower semicontinuous (by Mazur). The functional F(u) = integral |u|^q is convex and norm semicont. on L^2.

Separate proof that the \int | grad u |^2 is weakly lower semicont (use the variational characterization of the norm)

This material is in Reed-Simon, Appendix, Supplement to IV.5.

Types of converges in space of operators in Banach spaces. Norm, strong and weak operator convergence. Example. (Reed-Simon, Sec VI.1)

### Feb 4.

Banach and Hilbert space adjoints, basic properties. Reed-Simon Sec VI.1 and 2. Extra property mentioned: in H-space Ker T is the orthog. complement of Ran T*.

Definition of unitary, self adjoint, normal operators on Hilbert spaces.

Difference between symmetric and self-adjoint operators. Advertisement for the course Mathematical Physics by Professor Siedentop next semester.

A bounded operator T on a H-space is self-adjoint iff (Tx,x) is real for all x. Proof: via polarization exactly as in finite dimensional linear algebra. (Thm V.5.6 in Werner)

If T is selfadjoint, then the norm is sup_x |(Tx, x)| (Werner Thm V.5.7.)

Definition of resolvent, resolvent set, spectrum, eigenvalue, eigenvector, point spectrum, continuous spectrum, residual spectrum. (Werner VI.1 or Reed-Simon VI.3. Note that Reed-Simon does not define the continuous spectrum here, we use the terminology of Werner).

Motivation: Stable solution to equations Tx - lambda x = y.

The spectrum is the disjoint union of these three types of spectra. Proof: inverse mapping thm.

Next time we will need a few basic properties of analytic functions. Many students did not have complex analysis (Funktionentheorie). We will not have time to discuss it, but I wrote up a short Crash course on complex analysis. Please read it. If you never heard of these concepts and you find it too hard to understand them from this short write-up, then you will have to accept a few (sometimes surprising) facts along the proofs next week.

### Feb 8.

Example when the spectrum is not fully point spectrum.

Recall that in finite dimensions, the resolvent either does not exist because (lambda-T) is not injective (hence lambda is an eigenvalue), or if it is injective, then it is also surjective (count the dimension), hence the inverse exists, and in finite dimensions it is automatically bounded. So the spectrum is always point. In infinite dimensions the situation is more complicated.

Spectrum of T and its adjoint are the same (Thm. VI.1.2 Werner). Uses that if T is isometry iff T' is isometry (homework). For Hilbert space, the spectra are the conjugates of each other.

On a Hilbert space, the spectrum is in the closure of the set of all (Tx, x), |x|=1. Proof: Exercise (Lemma VII.1.1 Werner)

Spectrum of the selfadjoint operator is on the real line. In this case, there is no residual spectrum (use that Ker (lambda - T) is the orthog complement of Ran (lambda-T)^*)

Crash course on analytic functions. Analyticity, power series, radius of convergence, Laurent series, Liouville theorem. Look up Crash course on complex analysis

Banach space valued analytic functions.

Main theorem about the spectrum of bounded operators on a B space X:

(i) The resolvent set is open

(ii) The spectrum is not empty.

(iii) The resolvent is an analytic function from the resolvent set into the Banach space of bounded operators on X.

(iv) Resolvent identities.

### Feb 11.

Fredholm Alternative for matrices. Either the inverse of T exists or Tx=0 has solution. Moreover, Tx=y is solvable iff y is orthogonal to N(T^t).

Fredholm Alternative is wrong in inifite dimension. Finite dimensional proofs rely on the dimension formula (rank(T) + nullity(T) = n. Note: this is equivalent to ind(T)=0)

Compact operators, equivalent definitions: (i) image of a bounded set is precompact, i.e. the closure is compact; (ii) the image of every bounded sequence has a convergent subsequence.

A compact operator maps weakly convergent sequences into norm convergent ones.

Primary example: integral operators. E.g. K(x,y) continuous on [0,1]^2. It defines an operator on L^2[0,1] by (Kf)(x) = integral K(x,y)f(y) dy. This operator is compact. A similar proof gives that the operator is also compact in C[0,1]. (Proof: Exercise class)

Finite rank operators are compact. The set of compact operators is closed under norm limit.

In a H-space any compact operator can be norm-approximated by finite rank operators.

Fredholm alternative for operators T=I-A, A compact, on a Banach space: In this case, T has pseudoinverse, index(T)=0 and R(T) is closed.

The proof uses the Analytic Fredholm theorem in H-spaces (Thm VI.14 from RS), whose proof was sketched.

An application for f(z)=zA gives: either the inverse of (I-A) exists or Au=u has solution. This is a special case (for H spaces) of the full Fredholm alternative

Riesz Schauder Thm (VI.15 Reed-Simon): Compact operator on a H-space has discrete spectrum with the only (possible) accumulation point at zero. The multiplicities are finite.

Spectral theorem for compact operators on H. spaces. (without proof).(VI.16 Reed-Simon)