not mention it follows that all seminorms have the same zero set/kernel (it does not depend on ). So denote this common vector subspace by Given any the coset consists of all measurable functions that are equal to almost everywhere. Two cosets are equal if and only if almost everywhere, which happens if and only if The set of all cosets forms a vector space when vector addition and scalar multiplication are defined by and This space is the canonical quotient space of with respect to In the quotient space, two functions and are identified if (or equivalently, if ), which happens if and only if almost everywhere. The -norm on the quotient vector space Given any the value of the seminorm on the coset is constant and equal to denote this unique value by so that: This assignment defines a map, which will also be denoted by on the quotient vector space This map is a norm on called the . The value of a coset is independent of the particular function that was chosen to represent the coset since if is any other function then if and only if if and only if almost everywhere, in which case and so does indeed equal The Lebesgue space The normed vector space is called or the of -th power integrable functions and it is a Banach space for every (meaning that it is a complete metric space, a result that is sometimes called the Riesz–Fischer theorem). When the underlying measure space is understood then is often abbreviated or even just Depending on the author, the subscript notation might denote either or The above definitions generalize to Bochner spaces. In general, this process cannot be reversed: there is no consistent way to define a "canonical" representative of each coset of in For however, there is a theory of lifts enabling such recovery. Special cases Similar to the spaces, is the only Hilbert space among spaces. In the complex case, the inner product on is defined by The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in are sometimes called square-integrable functions, quadratically integrable functions or square-summable functions, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a Riemann integral . If we use complex-valued functions, the space