Topology/Hilbert Spaces

Topology Hilbert Spaces Banach Spaces Free group and presentation of a group A Hilbert space is a type of vector space that is complete and is of key use in functional analysis. It is a more specific kind of Banach space. Definition of Inner Product Space An inner product space or IPS is a vector space V over a field F with a function called an inner product that adheres to three axioms. 1. Conjugate symmetry: for all . Note that if the field is then we just have symmetry. 2. Linearity of the first entry: and for all and . 3. Positive definiteness: for all and iff . Definition of a Hilbert Space A Hilbert Space is an inner product space that is complete with respect to its inferred metric. Exercise Prove that an inner product has a naturally associated metric and so all IPSs are metric spaces. Example is a Hilbert space where its points are infinite sequences on I, the unit interval such that converges and is a Hilbert space with the inner product . Characterisation Theorem There is one separable Hilbert space up to homeomorphism and it is . Exercises (under construction) Topology Hilbert Spaces Banach Spaces Free group and presentation of a group