Hilbert Spaces: Orthonormal Systems

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Orthogonal Families and Projections

Introduces orthogonal families, orthonormal bases, and projections in linear algebra.

Function Spaces and Hilbert Spaces

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

Orthogonality and Least Squares Method

Covers orthogonal vectors, unit vectors, and the Pythagorean theorem in R^m.

The space of bandlimited functions

Explores the space of bandlimited functions, focusing on 1-bandlimited functions as a basis and their representation through sequences.

Vector spaces: Hilbert space

Introduces vector spaces, bases, and the Hilbert space, highlighting the practical implications of defining a basis in a vector space.

Orthogonal Projection: Example and Additional Remarks

Explains orthogonal projection onto a subspace and finding orthogonal bases using Gram-Schmidt procedure.

Linear Algebra: Quantum Mechanics

Explores the application of linear algebra in quantum mechanics, emphasizing vector spaces, Hilbert spaces, and the spectral theorem.

Finding Orthogonal/Orthonormal Base: First Step

Introduces the first step in finding an orthogonal/orthonormal base in a vector space.

Orthogonal Complement and Projection

Covers the concept of orthogonal complement and projection in vector spaces.

Projection in Vector Spaces

Explores the generalization of projection in vector spaces and its unique properties, emphasizing its role in finding the closest vector in a subspace.