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A function class is a set whose elements are functions, usually with a common domain of a function and codomain. For example,is a class of constant functions. A function class need not be a vector space: here $f_1+f_1$ has constant value two and lies outside $\mathcal{F}$.

An individual member is a function, or map; the class collects such members. This keeps statements about one transformation distinct from statements about a family of possible transformations.

Given a hypothesis class $\mathcal{H}$ and a loss function $\ell$, its loss class is the function classIt lets properties of a learning problem be studied through the random fluctuations of the losses themselves.

For independent and identically distributed random variables $Z_1,\dots ,Z_n$ and independent Rademacher signs ${\epsilon }_1,\dots ,{\epsilon }_n$, the Rademacher complexity of a function class $\mathcal{F}$ isIt measures how strongly the class can correlate with independent random signs on a random sample.

For an i.i.d. sample and an integrable function class $\mathcal{F}$,Introduce an independent ghost sample, replace each expectation by its ghost-sample average using Jensen inequality, and then multiply each paired difference by an independent Rademacher sign. Splitting the resulting supremum into its two sample contributions gives the factor two.