Content of the lecture
April 16th
Introduction; practical information; requirements for the
grade. Please sign
up for the Exercise class!
Motivation: Functional Analysis (FA) is 'infinite dimensional linear
algebra' - the study of infinte dimensional vector spaces and of
linear maps between them. It is a 'fusion' of Linear Algebra (LA) and
Analysis.
(From LA, the following
are assumed to be known: vector, matrix, linear (in)dependence, span,
basis, linear map and its matrix, range, kernel, rank, Gauss
elimination, scalar product, positive definite matrix, norm of a
vector and a matrix, orthonormal basis, orthogonal and unitary matrix,
orthogonal projection, change of basis formula.)
Both of these - LA and Analysis - are very much about studying
equations and their solutions (example given: linear systems,
eigenvalue eq, Intermediate Value Thm).
Lots of the motivation in FA
comes from studying equations - mostly differential and integral
eq's.
Example of general ODE (Gewoehnliche Diff Gleichung) and of 'integral
operators' (linear map K given by integrating (in t) function x(s) against a function k(t,s)
of two variables (s,t)).
Chapter 1: Topological and metric spaces.
Start on recalling Euclidean topology on R^n (distance, open sets).
April 18th
Continuation of recalling Euclidean topology on R^n (continuity, by epsilon-delta, and by open sets).
Definition of topological space, Hausdorff space. Remarks (discrete,
indiscrete topology). Definition of stronger and weaker
topologies. The discrete is stronger, the indiscrete weaker than any
other topology. Definition of relative/induced topology.
Definition of closed sets, the interior of a set, the closure of a
set. The interior is open, the closure is closed. Discussion of what
happens in discrete/indiscrete case.
Definition of neighbourhood of a point, inner point of a set, adherent
point of a set, limit point of a set, boundary points of a set,
boundary of a set.
Lemma: A set is open iff all its points are inner points. A set is
closed iff it contains all its adherent points. The interior of a set
is the set of all its inner points, and the closure of a set is the
union of the set and its limit points.
Definition of convergence of a sequence (discussion next time).
Definition of continuity of a map between two topological spaces.
April 23rd
Discussion of convergence of a sequences. Definition of open
map. Definition of homeomorphism, and homeomorphic top. spaces.
Definition of a neighbourhood basis of a point, and of first countable
top. space. Definition of sequentially continuous.
A continuous map f:X -> Y is seq. cont. IF X is first countable, and f
is seq. cont., then f is cont.
Let X be first countable, let A be subset of X, x adherent point of A. Then
there is a sequence of points _in_ A, which converges to x. In particular, the
closure of A equals the set of points which are limits of a sequence
of points from A.
Definition of local basis for x (a point). Definition of how to use
a local basis (one for _each_ x in X) to generate a topology.
April 25th
Definition of a topology generated by _any_ family of
subsets. Definition of subbasis and basis. Definition of second
countable. The Euclidean topology on R is second countable - the
family of open intervals with rational endpoints is a countable
basis.
Definition of a dense subset, and of a separable
top. space. Definition of the product topology on X x Y. Comments, in
particular: In the product topology, convergence is "coordinate
wise".
Definition of metric d, and metric space (X,d). Some further inequalities
(e.g. inverse triangle). Definition of the distance from a point to a
subset, and between two subsets of X.
Examples: Euclidean ('2-metric') on R^n, and the 'p-metrics'
on R^n, the discrete metric, the induced metric on a subset, the
French railroad metric. Definition of B_R(x), and K_R(x), of bounded
sets, and of the topology T_d generated by {B_{1/k}(x)} for k in
N.
Remark: The sets B_{1/k}(x) (and B_R(x)) are open. Any metric space is
first countable - hence, a map is continuous iff it is sequentially
continuous. Translation of convergence of sequences using the
metric. Note, the closure of B_R(x) _might_ not be K_R(x) ! The
metric (as a function of two variables) is continuous (proof next time).
April 30th
Proof of: The
metric (as a function of two variables) is continuous.
Definition of Cauchy sequence, and of a complete metric
space. Remark: Every convergent sequence is Cauchy, and every Cauchy
sequence is bounded.
Lemma: (X,d) complete. Subset A of X is a complete metric space iff
it is a closed subset of X.
Definition of isometry and of isometric metric spaces. Remarks.
Theorem: Completion of a metric space (unique up to isometry).
Discussion of fixpoint equations, and fixpoints. Definition of
Lipschitz continuous, Lipschitz constant, contraction. Definition of
uniformly continuous. Remarks: Lipschitz implies uniformly implies
continuous. Hence, contraction is continuous.
Formulation and
discussion of Banach's Fixpoint Theorem (proof next time).
May 2nd
Proof of Banach's Fixpoint Theorem. (Application to ODE's later.)
Definition of compact, sequentially compact, relative
compact subsets of a topological space. Discussion: In R^n compact equivalent to closed & bounded, in
general NOT.
Lemma: A continuous real map on a sequentially compact space attains
its minimum (and maximum).
If a space is first countable and compact, then it is sequentially
compact.
A compact subset of a Hausdorff space is closed. A closed subset of a
compact space is compact. Hence, in a compact Hausdorff space, subsets
are compact iff they are closed. (Proofs next time).
May 7th
Proof of: A compact subset of a Hausdorff space is closed. A closed subset of a
compact space is compact.
Proposition: A sequentially compact metric space is complete. (Lemma:
A Cauchy sequence with a convergent subsequence, is itself convergent,
with same limit).
Definition of pre-compact.
Theorem: Let (X,d) be complete metric space. A subset A is compact iff
it is sequentially compact. A _closed_ subset A is pre-compact iff it
is compact. (End of proof next time).
May 9th
End of proof from last time.
Theorem: Any compact metric space is separable. Theorem: X, Y
topological spaces, f: X-> Y map. If X is compact and if f is continuous,
then f(X) is compact (in Y). Corollary: A continuous real-valued
function on a compact topological space attains its maximum and
minimum. Discussion.
Theorem: (X,T) compact Hausdorff space. If T_w is any
topology strictly weaker than T, then (X,T_w) is not Hausdorff. If T_s
is any topology strictly stronger than T, then (X,T_s) is not compact.
For more on general topology, see for example
W. A. Sutherland: Introduction
to metric and topological spaces, Oxford University
Press.
J. L. Kelley: General
Topology, Springer Verlag.
G. K. Pedersen: Analysis
Now, Springer Verlag.
Chapter 2: Banach and Hilbert spaces.
Definition of norm, normed space, semi-norm. Remarks: Definition of
metric given by norm. The norm is continuous (from X to R) (with respect to topology
given by metric). Any convergent sequence is bounded. The linear
structure (vector addition and scalar multiplication) is
continuous. The balls {B_{1/k}(0)}_{k in N} form a countable
neighbourhood basis of 0 (the zero-vector in X). One gets a countable
neighbourhood basis of any x in X by translation: B_R(x) = x +
B_R(0).
Definition: A complete normed space is called a Banach space.
Examples: R^n with usual Euclidean norm. R^n with p-norm. (Continued
next time).
May 14th
Example: l_p, with p-norm (p in [1,infinity]), is a normed space. It is a Banach space.
Definition: Equivalent norms on a normed space. Example: Different l_p - norms on R^n.
Theorem: Any two norms on a finite dimensional normed vector space are
equivalent. Any finite dimensional normed vector space is a Banach
space. Any finite dimensional linear subspace of a normed vector space
is a closed subset, and is complete (even if larger space is not).
Riesz' Lemma: X Banach space, U prober closed linear subspace, lampda
in (0,1). There exists vector x of norm 1, such that ||x-u|| >= lambda
for all u in U. (Proof next time.)
May 15th
Proof of Riesz' Lemma. (NOTE: The space need NOT to be Banach - can
be left out in assumption - it was assumed in lecture).
Theorem: The closed unit ball in a normed space is compact iff the space is finite dimensional. (NOTE: The space need NOT to be Banach - can
be left out in assumption - it was assumed in lecture).
Definition of linear maps between normed spaces, and of their kernel
and range (which are linear subspaces). The kernel is closed if the
map is continuous. Theorem: A linear map is continuous iff it is
continuous at 0 iff it is a bounded linear operator (there exists C>0
sucht that ||Tx|| <= C ||x|| ). Definition of operator "norm", and of
B(X,Y). Lemma on facts on the operator "norm".
Any linear map between two finite dimensional normed spaces is
bounded/continuous.
Theorem: B(X,Y) with the operator "norm" is a normed space (ie, the operator
"norm" is a norm). If Y is Banach, then so is
B(X,Y) with the operator norm. (Rest of proof
next time).
May 21st
Rest of proof from last time (see above).
Definition of the dual space of a normed vector space and its
norm. The dual space is always a Banach space (with the operator
norm).
Examples: The dual of any finite dimensional normed space is
(isometrically isomorpic to) itself.
The dual of l_p (1 < p < infinity)
is (isometrically isomorphic to) l_q (q the conjugate exponent) (including proof).
Theorem: X, Y normed spaces, Y Banach, M dense linear subspace of X, T:M -> Y linear and bounded. Then T has a unique extension to
a linear bounded map T-tilde:X -> Y, with the same operator norm (proof: next time).
May 23rd
Proof from last time (see above).
Definition of a compact map between normed spaces, and of a compact
operator, and of K(X,Y). A compact operator is bounded, so K(X,Y)
subset B(X,Y). Definition of continuous, compact, and dense
embeddings. Discussion. The composition of a compact and a bounded
operator is again compact.
Definition of inner product/scalar product, inner product
space. Remarks. Cauchy-Schwarz-Bunyakowski's inequality.
The inner product generates a norm. The inner product is continuous in
this norm. The Parallelogram Identity. Definition of a Hilbert space.
May 30th
Examples: R^n,C^n (usual scalar products), R^n with _any_ scalar
product, l_2, C[0,1] with < f,g >=int \overline{f(x)} g(x) dx (NOT
Hilbert space).
The scalar product with a fixed vector y defines a bounded linear
functional. Conversely: Riesz' representation theorem: Every bounded
linear functional f on a Hilbert space is given in this way. The
vector y also minimizes F(z)= < z,z > - 2 Re f(z).
Definition of orthogonal vectors, and orthogonal complement of a set.
The Projection Theorem (proof next time).
June 4th
Proof of the projection theorem (from last time).
Definition of boundedness and coercivity of a sesqui linear form on a normed space.
Lax-Milgram's Theorem: For any bounded, coercive sesquilinear form b on a Hilbert space there is a bounded linear bijection R such
that b(Rx,y) = < x, y > for all x,y in H.
Discussion of algebraic basis for a vector space: Definition of
algebraic (Hamel) basis. Theorem: Every vector space has a Hamel
basis. Proof follows from Zorn's Lemma. Proposition: Let X be a Banach
space which is NOT finite dimensional, and let B be any Hamel basis
for X. Then B is
uncountable. This makes the concept of a Hamel basis useless (for us!).
June 5th
For general index set I, and Banach space X: Defintion of family x:I
-> X, and of absolutely summable families. Definition of support of an
absolutely summable family (a countable subset of I). Definition of
square absolutely summable families, and of the Hilbert space
l_2(I).
Definition of an orthonormal systen (ONS) and a maximal ONS, in inner
product space. Examples. Definition of Fourier-coefficients x^(i) of an
element x in H with respect to an ONS.
Lemma: Pythagoras. Bessel's Inequality.
Given an ONS (indexed over I) for H, and an element x in l_2(I), definition
of sum x_i e_i, as element in H.
Definition of the Fourier-map, taking x in H into the family of its
Fourier coefficients (an element of l_2(I)). It is linear, bounded, surjective, with operator norm less
or equal 1.
Theorem: An ONS is maximal iff the Fourier map is injective iff the
Fourier map is an isometry iff the linear span of the ONS is dense
in H iff Parseval's Identity holds iff for all x, the element sum x^(i) e_i
equals x. (Proof: Next time).
June 11th
Proof of the equivalent characterizations of an ONB. Every Hilbert space
has an ONB and is isometrically isomorphic to l_2(I), but this is
constructive only if the space is separable.
Chapter 3: (Some) function spaces.
The space C(X) of continuous functions over a compact metric space X.
Theorem: C(X) equipped with the supremum norm is a Banach space. The
proof is a "3epsilon argument" (useful trick).
What could prevent us from selecting a convergent subsequence from a
family in C(X)? Essentially two mechanisms: the sequence spikes up to
infinity or goes wild oscillating to death. Then if we constrain the
family to be bounded and equicontinuous it is expected that a convergent
subsequence do exist. This is precisely Ascoli-Arzela'.
Theorem (Ascoli-Arzela'): a bounded and equicontinuous subset of C(X) is
relatively compact. In the equivalent language of sequential compactness
a uniformly bdd and equicont. family of continuous functions on X admits
a subsequence that converges uniformly. The proof is a "diagonal
trick" (another useful trick).
[[ folklore: Ascoli came first and proved the sufficient condition for
pre-compactness. Arzela' then showed that it is also necessary. This is
why French and Italian maths literature report Ascoli-Arzela'. Anglo
Saxon literature adopts instead the alphabetical order. ]]
Hoelder spaces. These are spaces of functions with some prescribed
regularity. Program: need first to formalize their domain, as well as
the notion of regularity. Then the norm-space structure and their
completeness.
Definition of: domain in R^n, compactly contained subset of a domain,
support of a C(X)-function.
Definition of C^m and C^infinity functions on a domain Omega, and their
counterpart with compact support. Definition of spaces of bounded and
uniformly cont. functions on a domain, with their natural norm. A useful
notation enters the def: multi-indices.
June 13th
Definition of Hoelder continuous functions with exponent gamma
(Lipschitz when gamma=1) with their natural norm. Notice: such a norm
captures both the sup of the function and all its derivatives, and the
smallest possible incremental ratio with exponent gamma.
Theorem: Hoelder cont. functions form a Banach space. The proof of
completeness has the usual scheme: we pick a Cauchy sequence and we show
that it converges. Since it is Cauchy also uniformly, we first show that
the sequence and all its derivatives converge uniformly. This gives us a
pointwise convergence to control the incremental ratio.
If the domain is bounded we can claim something more:
Theorem: the spaces of gamma-Hoelder continuous functions embed compactly into
each other when gamma increases. (They are nested, the larger the gamma
the smaller the space.)
Overview of L^p spaces. We shall make systematic reference to Prof.
Mueller's Analysis III lecture as well as to Prof. Griesemer's survey
notes. Notions assumed to be well-known: sigma-algebras, measurable
functions, def of integration. In the following we shall recall the L^p
theory without proofs.
Definition of: measurability and integrability of a complex-valued
function, p-seminorms for p between 1 and infinity, endpoint included.
The case p=infinity involves the essential supremum.
Theorem: the p-seminorms satisfy the generalised Hoelder as well as the
Minkowski inequality. If the measure space has finite measure, the
q-seminorm controls the p-seminorm if q is larger than p. Therefore the
embedding is continuous. The finiteness of the measure of the measure
space is clearly essential.
Taking equivalence classes with respect to equality of functions almost
everywhere, we turn the p-seminorm into a norm. We thus define the
normed vector spaces L^p. Rigorously speaking they consists of
equivalence classes. We will allow ourselves a standard abuse of
notation.
Definition: absolutely continuous measure. Radon-Nikodyn derivative (or
density) of an absolutely continuous measure.
Theorem of Radon-Nikodyn.
June 18th
If mu(X) < infinity, and p < q, then L^q is dense in L^p (in p-norm).
Riesz-Fischer: L^p with p-norm is a Banach space. In particular, L^2
with the scalar product < f,g > = int \overline{f} g is a Hilbert
space.
For all p (also p = infinity) and all k in N, C_0^infinity(R^n) subset C_0^k(R^n)
subset C_0(R^n) subset L^p(R^n) (with
Lebesgue measure). If p < infinity, then
C_0^infinity(R^n) is dense in
L^p(R^n). (This is NOT the case for
p=infinity.) L^p(R^n) is separable.
(Repetition:
Definition: absolutely continuous measure. Radon-Nikodyn derivative (or
density) of an absolutely continuous measure.
Theorem of Radon-Nikodyn.)
Dual of L^p(X,mu) is isometrically isomorphic to L^q(X,mu), with q the
conjugate exponent WHEN 1<= p < infinity AND (X,mu) is
sigma-finite. That is, any bounded
linear functional on L^p is simply
integration against some
L^q-function. (End of proof next time).
June 20th
End of proof from last time.
Remark on PDE, regularity, the motivation for Hoelder-spaces, and why
sometimes not good enough. Motivation for weak derivatives.
Definition of weak derivatives. Weak derivatives are unique, and, if
classical derivative exist, they coincide (as L^1_loc
functions!). Uses the Fundamental Lemma of Calculus of Variations (not
proved).
Definition of Sobolev spaces, and their norms.
June 25th
Completeness of Sobolev spaces W^{m,p}. W^{m,2} are Hilbert spaces. Remarks.
For bounded domains: The smooth Sobolev functions (intersection smooth
with W^{m,p}) form a dense subset
of W^{m,p}
(independent of the regularity of the boundary of the domain).
Definition of regularity of domains (rather, their boundary). For
C^0-domains, the restriction of C_0^infinity(R^n) - functions is dense
in W^{m,p}.
Sobolev embeddings: W^{m,p} embed continuously into L^{np/(n-mp)} when
1 <= mp < n (_bounded_ domain). W^{1,p} embed compactly into L^q for
any q < np/(n-p) (also _bounded_ domain).
Remark on PDE, integral operators, choice of spaces/norms etc.
Chapter 4: The cornerstones of Functional Analysis
Definition of no-where dense and meagre sets, and of sets of first and
second category.
Baire's category theorem: The intersection of countably many open and
dense subsets of a complete metric space is itself dense. (End of
proof next time).
June 27th
End of proof of Baire's Category Theorem. Corollary: A complete metric
space is of second category (in itself). Corollary: In a complete
metric space, if the union of
countably many closed sets contains an open ball, then at least one of
the sets must contain an open ball too.
Principle of Uniform Boundedness / Banach - Steinhaus Theorem: For X
Banach, Y normed, and a subset H of B(X,Y) which is _pointwise_
bounded, H is in fact uniformly bounded. Corollary: The pointwise
limit of a sequence of bounded linear operators is a bounded linear
operator.
Open Mapping Theorem / Inverse Mapping Theorem: X, Y Banach, T in
B(X,Y) surjectiv. Then T is an open map. In particular, if T is a
bijection, then T^{-1} is bounded (ie in B(Y,X)). (Re-cap and end of
proof next time).
July 2nd
Re-cap and end of proof of Open Mapping Theorem. Corollary: If two
norms on a space X both makes X a Banach space, and if one is bounded
by the other, then they are equivalent.
Definition of graph G(T) of a map T, and of norm on X x Y for X, Y normed
spaces. If X, Y are Banach, so is X x Y (seen earlier). Closed Graph
Theorem: A linear map T:X -> Y (X, Y Banach) is bounded iff G(T) is closed (in X x Y). Discussion of the connection
and difference between proving "T continuous" and "G(T) closed".
Definition of sublinear functionals.
Hahn-Banach Theorem (extension version): X a _real_ vector space, p a
sublinear functional on X, M linear subspace of X, f linear
functional on M, bounded pointwise (on M) by p. Then there
exists an extension F (of f) to X, F a linear map, bounded
by p (on X).(End of proof next time).
July 4th
End of proof last time (applying Zorn's Lemma).
Theorem: X K-vector space, M linear subspace, p semi-norm on X, f
linear map on M, with |f| =< p on M. Then there exists extension F (of
f) to X, F linear, with |F|
<= p.
Corollary: X normed K-vector space, M linear subspace, f:M -> K linear
and bounded. Then there exists F in X'
which extends f, and the norms
||f||_{B(M,K)} and ||F||_{B(X,K)} are equal.
Theorem (Hahn-Banach separation theorem): X normed K-vector space, A,B
disjoint convex subsets of X, A
open. Then there exist F in X', and
real number gamma so that
Re F(x) < gamma <= Re F(y) for all x in A, y in B. Discussion: The
set where F(z) = gamma (an affine
subspace of co-dimension 1) separates A
and B.
July 9th
End of proof of Hahn-Banach separation theorem.
Theorem: X Banach, M closed linear subspace, x not in M. Then there
exists F in X' with norm 1, value at x equal dist(x,M)>0, and F
restricted to M is zero. Consequence: For all non-zero
vectors x in a Banach space, there is an element in X' with
norm one such that F(x) = ||x||. Therefore, X' separates
points in X.
Definition of bi-dual X'' of normed space X. The bi-dual is always
Banach. Examples. Definition of canonical embedding:
isometric embedding of X into X'' (via X', using H-B). Discussion
of completion of non-Banach X via X'' and canonical
embedding.
Definition: weakly bounded. Proposition: Weakly bounded implies
norm-bounded (via Banach-Steinhaus).
Definition: Reflexiv normed space. Examples and discussion. Definition
of weak convergence (in normed space) and weak-*
convergence (in a dual space), of weakly Cauchy, weak-*
Cauchy, and of weakly sequentially compact and weak-*
sequentially compact subsets. Discussion of difference
between weak and weak-* convergence on the dual space X'
(these two are the same if X is reflexive space).
July 11th
Recall: Definition of weak and weak-* convergence. Definition of the
'weak topology' (the topology giving rise to weak
convergence). Discussion. 'Usual' (norm) convergence is called strong
convergence.
Remarks: The weak limit is unique (via Hahn-Banach). So is the weak-*
limit. Strong convergence imply weak convergence. The opposite is not
true (take ONB in Hilbert space). If X is reflexive, V closed linear
subspace, then also V is reflexive. X is reflexive iff X' is
reflexive. If X' is separable, then X is separable (converse not
true). The norm is lower (sequentially) semi-continous with respect to
weak convergence, as well as with respect to weak-*
convergence. Discussion: Importance for minimization. Weak-convergent,
and weak-* convergent sequences are norm-bounded.
Theorem: X separable. Then the closed unit ball in X' is weak-*
sequentially compact (ie, any bounded sequence in X' has a weak-*
convergent subsequence).
Theorem (Banach-Alaoglu): X reflexive Banach space. Then any (norm-)
bounded sequence in X has a weakly convergent subsequence (ie. the
closed unit ball in X is weakly sequentially compact).
Chapter 5: Topics on bounded operators.
(This chapter contains less (!) proofs)
Recall: Definition of compact operators.
July 16th
Favorite examples of compact operators: compact embeddings; certain integral operators.
Definition of (Banach space) adjoint of a bounded operator. Definition
of Hilbert space adjoint of bounded operator. Discussions, and
algebraic properties. The inverse of a bounded exists iff the inverse
of the adjoint exists.
Definition of resolvent set, spectrum, point spectrum, continuous
spectrum, rest spectrum for bounded operator on a Banach space (T in
B(X)). Discussions and remarks. Definition of eigenvalue (point
spectrum) and eigenvector, and discussion of relation
(similarities/differences) to finite
dimensional case.
Definition of resolvent map. The resolvent map is a complex analytic
map on the resolvent set (with values in B(X)).
Thm: If X Banach, T in B(X) with norm < 1, then I-T is invertible, and
it is given by a convergent
power series (the Neumann
series) a la for the geometric
series. Consequence: The set
of invertible operators is an
open subset of
B(X,Y).
Definition of f(T) for T in B(X) and f monomial, polynomial,
convergent (complex, complex
valued) power series. Examples
and discussion.
July 19th
Discussion of solutions to (linear) equations Tx - lambda x = y:
Number of 'degrees of freedom' (possible number of independent
solutions) and number of 'constraints'. Motivates: Definition of
Fredholm operator and its index. Theorem: For T compact, I - T is
Fredholm, with index zero. In particular, if I - T is injective, then
it is surjective (no proof).
Theorem: Spectral Theorem for Compact operators (on complex Banach
space) (Riesz-Schauder). Statement and discussion (no proof).
Corollary: For a compact operator T, the resolvent map has an isolated
pole at the (isolated) non-zero eigenvalues. The order of
the pole is the order of the eigenvalue.
Theorem (Fredholm alternative): For T compact, either Tx - lambda x = y has
unique solution x for all y, or the equation Tx - lambda x =0 has
non-trivial solutions (no proof).
Theorem (Fredholm): T compact on Banach space. Then Tx - lambda x =
y has solution x, for y (fixed) iff x'(y) = 0 for all solutions x' in
X' to T'x' - lambda x' = 0. Furthermore, the number of constraints on
y is equal to the number of linearly independent solutions to Tz -
lambda z=0 (no proof).
Theorem (Schauder): T in B(X,Y) is compact iff T' in B(Y',X') is
compact (no proof).
Eksempel: 'Diagonal' operator on a Hilbert space (sum of weighted
projections). The operator is normal (TT^* = T^*T). If the weights go
to zero, the operator is
compact.
Spectral Theorem for compact, normal operators on a Hilbert space (no
proof): Any such operator is of the form in the example.
Last update: July 20th, 2012 by Thomas Østergaard Sørensen.