Episode 1 Inner product space and measure theory Msc 4th sem define Pre Hilbert space with examples

Описание: Inner Product Spaces and Measure Theory by Yogeshwar Sir @yogeshwarmaths85
Anant academy of mathematics
#MSc_ maths
#MSc_mathematics
#Yogeshwarsir

Email: [email protected]
Paper Code : 21MAT24C1
M. Marks = 100
Term End Examination = 80
Assignment = 20
Time = 3 Hours
Course Outcomes
Students would be able to:
CO1 Understand #Hilbert spaces and related terms.
CO2 Introduce the #concept of #projections, #measure and #outermeasure.
CO3 Learn about #Hahn, #Jordan and #Radon-#Nikodyn #decomposition #theorem, #Lebesguestieltjes #integral,
#Baire #sets and #Baire #measure
Section - I
#Hilbert #Spaces: #Inner #product spaces, Hilbert spaces, #Schwarz #inequality, #Hilbert space as #normed #linear
#space,#Convex sets in Hilbert spaces, #Projection theorem, #Orthonormal sets, #Separability, Total
#Orthonormal sets, #Bessel inequality, #Parseval #identity.
Section - II
#Conjugate of a Hilbert space, #Riesz representation theorem in Hilbert spaces, Adjoint of an operator on a
#Hilbert space, #Reflexivity of #Hilbert space, #Self-adjoint #operators, #Positive #operators, #Product of #Positive
#Operators.
Section - III
#Projection #operators, #Product of #Projections, #Sum and #Difference of #Projections, #Normal and #unitary
#operators, #Projections on Hilbert space, #Spectral theorem on #finite #dimensional space. #Convex #functions,
#Jensen inequalities, #Measure space, #Generalized Fatou lemma, Measure and outer measure, #Extension of
a measure, #Caratheodory #extension #theorem.
Section - IV
#Signed measure, #Hahn #decomposition theorem, #Jordan #decomposition theorem, #Mutually #signed
measure, #Radon – Nikodyn theorem,#Lebesgue #decomposition, #Lebesgue - Stieltjes integral, Product
#measures, #Fubini theorem, #Baire sets, #Baire measure, #Continuous functions with #compact support.