Lecture 4: Hilbert Spaces and Unique Solutions

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Introduces continuity in function spaces and the Galerkin method for solving boundary value problems.

Covers the definition and properties of Hilbert spaces, including the Cauchy-Schwarz inequality and norm definition.

Introduces function spaces and Hilbert spaces, discussing inner product spaces and the importance of completeness in Hilbert spaces.

Covers normed spaces, dual spaces, Banach spaces, Hilbert spaces, weak and strong convergence, reflexive spaces, and the Hahn-Banach theorem.

Covers distributions, derivatives, convergence, and continuity criteria in function spaces.

Covers normed spaces, Banach spaces, and Hilbert spaces, as well as dual spaces and weak convergence.

Delves into the completeness of Lp spaces and the density of function classes in L².

Explores Sobolev spaces in higher dimensions, discussing derivatives, properties, and challenges with continuity.

Covers the states of composite systems in Hilbert space, including operators, observables, tensor products, eigenvalues, and partial measurements.

Explores convolution operators, interpolation spaces, and function convergence in different spaces.