18.102 - Introduction to Functional Analysis
18.102 Introduction to Functional Analysis (Spring 2021, MIT OCW). Instructor: Dr. Casey Rodriguez. Functional analysis helps us study and solve both linear and nonlinear problems posed on a normed space that is no longer finite-dimensional, a situation that arises very naturally in many concrete problems. For example, a nonrelativistic quantum particle confined to a region in space can be modeled using a complex valued function (a wave function), an infinite dimensional object (the function's value is required for each of the infinitely many points in the region). Functional analysis yields the mathematically and physically interesting fact that the (time independent) state of the particle can always be described as a (possibly infinite) superposition of elementary wave functions (bound states) that form a discrete set and can be ordered to have increasing energies tending to infinity. The fundamental topics from functional analysis covered in this course include normed spaces, completeness, functionals, the Hahn-Banach Theorem, duality, operators; Lebesgue measure, measurable functions, integrability, completeness of Lp spaces; Hilbert spaces; compact and self-adjoint operators; and the Spectral Theorem. (from ocw.mit.edu)
| Lecture 18 - The Adjoint of a Bounded Linear Operator on a Hilbert Space |
Instructor: Dr. Casey Rodriguez. We define adjoint operators via the Riesz Representation theorem. This allows us to prove an analogue of the Rank-Nullity theorem for bounded linear operators. Then, we start exploring compact operators a little, which we will soon see are " almost matrices ".
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