In mathematics, and especially in harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes L^p spaces. Like ${\displaystyle L^{p}}$ spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932.

Besides ${\displaystyle L^{p}}$ spaces, a variety of function spaces arising naturally in analysis are Orlicz spaces. One such space is ${\displaystyle L\,\log ^{+}\!L}$, which arises in the study of Hardy–Littlewood maximal functions, consisting of measurable functions ${\displaystyle f}$ such that

Here ${\displaystyle \log ^{+}}$ is the positive part of the logarithm. Also included in the class of Orlicz spaces are many of the most important Sobolev spaces. In addition, the Orlicz sequence spaces are examples of Orlicz spaces.

These spaces are called Orlicz spaces because Władysław Orlicz was the first who introduced them, in 1932. Some mathematicians, including Wojbor Woyczyński, Edwin Hewitt and Vladimir Mazya, include the name of Zygmunt Birnbaum as well, referring to his earlier joint work with Władysław Orlicz. However in the Birnbaum–Orlicz paper the Orlicz space is not introduced, neither explicitly nor implicitly, hence the name Orlicz space is preferred. By the same reasons this convention has been also openly criticized by another mathematician (and an expert in the history of Orlicz spaces), Lech Maligranda. Orlicz was confirmed as the person who introduced Orlicz spaces already by Stefan Banach in his 1932 monograph.

Let ${\displaystyle \mu }$ be a σ-finite measure on a set ${\displaystyle X}$, and ${\displaystyle \Phi :[0,\infty )\to [0,\infty ]}$ a Young function; i.e., a convex, lower semicontinuous, and non-trivial function. Non-trivial in the sense that it is neither the zero function ${\displaystyle x\mapsto 0}$ nor the convex dual of the zero function

Now let ${\displaystyle L_{\Phi }^{\dagger }}$ be the set of measurable functions ${\displaystyle f:X\to \mathbb {R} }$ such that the integral

is finite, where, as usual, functions that agree almost everywhere are identified.

This is not necessarily a vector space (for example, it might fail to be closed under scalar multiplication). The Orlicz space, denoted ${\displaystyle L_{\Phi }}$, is the vector space of functions spanned by ${\displaystyle L_{\Phi }^{\dagger }}$; that is, the smallest linear space containing ${\displaystyle L_{\Phi }^{\dagger }}$. Formally,