‖·‖ Functional Analysis

Сургамж

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.

Сургамж

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

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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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