Elements of Linear Algebra: Vector spaces, linear operators, convex sets.
Norms in Vector Spaces: Fundamental definitions, balls in normed spaces, the interrelation between algebraic and topological structures of normed spaces, Banach spaces.
Continuous or Bounded Operators: Basic properties, the norm in the space
B
(
X
,
Y
)
B(X,Y), the space
X
∗
X
∗
of linear functionals defined on
X
X, isomorphisms and isometries between normed spaces, subsequent continuity of linear operators in finite-dimensional spaces.
Hilbert Spaces: Inner product, the dual space of a Hilbert space, orthonormal systems.
Hahn-Banach Theorem: Consequences of the theorem, natural imbedding of
X
X into
X
∗
∗
X
∗∗
, the dual spaces of
L
p
L
p
.
Geometrical Form of Hahn-Banach Theorem: Minkowski functional, Hahn-Banach separation theorems, the theorem of Krein-Milman.
Applications of Baire’s Theorem in Banach Spaces: Uniform boundedness principle, the open mapping theorem, the closed graph theorem, quotient spaces.
- Teacher: Νικόλαος Γιαννακάκης
ECTS : 6
Language : el