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Functional Analysis

Linear algebra in infinitely many dimensions, where functions are the vectors. Quantum mechanics, PDEs and signal processing all live here; the spectral theorem is the eigenvalue story told for operators.

经验教训

Advanced Normed and Banach spaces Vector spaces with a length; completeness; the classical sequence and function spaces. sqrt(3^2 + 4^2) Advanced Hilbert spaces and the spectral theorem Inner products, orthonormal bases, Fourier series, and eigenvalues of operators. eigenvalues of [[2,1],[1,2]]

Symbols used here

\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.
\lambda
lambda (eigenvalue)
The factor by which an eigenvector is stretched: Av = λv.
\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|
dot product, norm
Σ u_i v_i; the length of v, √(v·v).
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
\|f\|_p,\ \langle f, g \rangle
p-norm, inner product
Length of a function; the generalised dot product.

Questions people ask

What is a Hilbert space?

A vector space with an inner product (so lengths and angles make sense) that is complete (no missing limit points). Square-integrable functions form one; quantum states live in one.

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