Normalizable

Normalizable describes an object that can be scaled to a standard size, typically unit integral or unit norm. In analysis, a function f on a measure space is normalizable if the integral of its absolute value is finite and nonzero, so that the normalized function f* = f / ∫ |f| dμ has total integral 1. In quantum mechanics, a state ψ is normalizable if its norm ∥ψ∥ is finite; then it can be rescaled by dividing by ∥ψ∥ to obtain a unit-norm state.

For probability densities, a nonnegative function p with finite integral can be turned into a density by dividing by its integral. A Gaussian function is normalizable; many localized, square-integrable functions are normalizable. By contrast, a plane wave e^{ikx} on the entire real line is not normalizable, since the integral of |e^{ikx}|^2 over the infinite line diverges.

Normalizing constant: If f is normalizable, the constant C = 1 / ∫ |f| dμ (or C = 1 / ∑ |a_n| for a sequence) yields the normalized version f_norm = C f, ensuring unit total mass or unit norm. In discrete probability, this is equivalent to defining a probability distribution from an unnormalized weight by dividing by the sum of weights.

In summary, normalizability concerns the possibility of scaling an object so that it satisfies a standard size criterion, enabling interpretation as a probability density, a quantum state, or a unit-norm element of a function space.