Lp space
Special cases

Similar to the ℓ p {\displaystyle \ell ^{p}} spaces, L 2 {\displaystyle L^{2}} is the only Hilbert space among L p {\displaystyle L^{p}} spaces. In the complex case, the inner product on L 2 {\displaystyle L^{2}} is defined by ⟨ f, g ⟩ = ∫ S f (x) g (x) ¯ d μ (x) {\displaystyle \langle f,g\rangle =\int _{S}f(x){\overline {g(x)}}\,\mathrm {d} \mu (x)}

The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in L 2 {\displaystyle L^{2}} 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 (Titchmarsh 1976).

If we use complex-valued functions, the space L ∞ {\displaystyle L^{\infty }} is a commutative C*-algebra with pointwise multiplication and conjugation. For many measure spaces, including all sigma-finite ones, it is in fact a commutative von Neumann algebra. An element of L ∞ {\displaystyle L^{\infty }} defines a bounded operator on any L p {\displaystyle L^{p}} space by multiplication.

For 1 ≤ p ≤ ∞ {\displaystyle 1\leq p\leq \infty } the ℓ p {\displaystyle \ell ^{p}} spaces are a special case of L p {\displaystyle L^{p}} spaces, when S = N) {\displaystyle S=\mathbb {N})} consists of the natural numbers and μ {\displaystyle \mu } is the counting measure on N. {\displaystyle \mathbb {N}.} More generally, if one considers any set S {\displaystyle S} with the counting measure, the resulting L p {\displaystyle L^{p}} space is denoted ℓ p (S). {\displaystyle \ell ^{p}(S).} For example, the space ℓ p (Z) {\displaystyle \ell ^{p}(\mathbb {Z})} is the space of all sequences indexed by the integers, and when defining the p {\displaystyle p} -norm on such a space, one sums over all the integers. The space ℓ p (n), {\displaystyle \ell ^{p}(n),} where n {\displaystyle n} is the set with n {\displaystyle n} elements, is R n {\displaystyle \mathbb {R} ^{n}} with its p {\displaystyle p} -norm as defined above. As any Hilbert space, every space L 2 {\displaystyle L^{2}} is linearly isometric to a suitable ℓ 2 (I), {\displaystyle \ell ^{2}(I),} where the cardinality of the set I {\displaystyle I} is the cardinality of an arbitrary Hilbertian basis for this particular L 2. {\displaystyle L^{2}.}
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