Lp space

In mathematics, the L spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910).

L spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines.

The Euclidean length of a vector \(x = (x_1, x_2, \dots, x_n)\) in the \(n\)-dimensional real vector space \(\Reals^n\) is given by the Euclidean norm: \[\|x\|_2 = \left({x_1}^2 + {x_2}^2 + \dotsb + {x_n}^2\right)^{1/2}.\]

The Euclidean distance between two points \(x\) and \(y\) is the length \(\|x - y\|_2\) of the straight line between the two points. In many situations, the Euclidean distance is appropriate for capturing the actual distances in a given space. In contrast, consider taxi drivers in a grid street plan who should measure distance not in terms of the length of the straight line to their destination, but in terms of the rectilinear distance, which takes into account that streets are either orthogonal or parallel to each other. The class of \(p\)-norms generalizes these two examples and has an abundance of applications in many parts of mathematics, physics, and computer science.

For a real number \(p \geq 1,\) the \(p\)-norm or \(L^p\)-norm of \(x\) is defined by \[\|x\|_p = \left(|x_1|^p + |x_2|^p + \dotsb + |x_n|^p\right)^{1/p}.\] The absolute value bars can be dropped when \(p\) is an even integer, and \(x\) is drawn from the set of real numbers, or one of its subsets.

The Euclidean norm from above falls into this class and is the \(2\)-norm, and the \(1\)-norm is the norm that corresponds to the rectilinear distance.

The \(L^\infty\)-norm or maximum norm (or uniform norm) is the limit of the \(L^p\)-norms for \(p \to \infty\), given by: \[\|x\|_\infty = \max \left\{|x_1|, |x_2|, \dotsc, |x_n|\right\}\]

For all \(p \geq 1,\) the \(p\)-norms and maximum norm satisfy the properties of a "length function" (or norm), that is:

• only the zero vector has zero length,
• the length of the vector is positive homogeneous with respect to multiplication by a scalar (positive homogeneity), and
• the length of the sum of two vectors is no larger than the sum of lengths of the vectors (triangle inequality).

Abstractly speaking, this means that \(\Reals^n\) together with the \(p\)-norm is a normed vector space. Moreover, it turns out that this space is complete, thus making it a Banach space.

The \(p\)-norm can be extended to vectors that have an infinite number of components (sequences), which yields the space \(\ell^p.\) This contains as special cases:

• \(\ell^1,\) the space of sequences whose series are absolutely convergent,
• \(\ell^2,\) the space of square-summable sequences, which is a Hilbert space, and
• \(\ell^\infty,\) the space of bounded sequences.

The space of sequences has a natural vector space structure by applying scalar addition and multiplication. Explicitly, the vector sum and the scalar action for infinite sequences of real (or complex) numbers are given by: \[\begin{aligned} & (x_1, x_2, \ldots, x_n, x_{n+1},\ldots)+(y_1, y_2, \ldots, y_n, y_{n+1},\ldots) \\ = {} & (x_1+y_1, x_2+y_2, \ldots, x_n+y_n, x_{n+1}+y_{n+1},\ldots), \\[6pt] & \lambda \cdot \left (x_1, x_2, \ldots, x_n, x_{n+1},\ldots \right) \\ = {} & (\lambda x_1, \lambda x_2, \ldots, \lambda x_n, \lambda x_{n+1},\ldots). \end{aligned}\]

Define the \(p\)-norm: \[\|x\|_p = \left(|x_1|^p + |x_2|^p + \cdots +|x_n|^p + |x_{n+1}|^p + \cdots\right)^{1/p}\]

Here, a complication arises, namely that the series on the right is not always convergent, so for example, the sequence made up of only ones, \((1, 1, 1, \ldots),\) will have an infinite \(p\)-norm for \(1 \leq p < \infty.\) The space \(\ell^p\) is then defined as the set of all infinite sequences of real (or complex) numbers such that the \(p\)-norm is finite.

One can check that as \(p\) increases, the set \(\ell^p\) grows larger. For example, the sequence \[\left(1, \frac{1}{2}, \ldots, \frac{1}{n}, \frac{1}{n+1}, \ldots\right)\] is not in \(\ell^1,\) but it is in \(\ell^p\) for \(p > 1,\) as the series \[1^p + \frac{1}{2^p} + \cdots + \frac{1}{n^p} + \frac{1}{(n+1)^p} + \cdots,\] diverges for \(p = 1\) (the harmonic series), but is convergent for \(p > 1.\)

One also defines the \(\infty\)-norm using the supremum: \[\|x\|_\infty = \sup(|x_1|, |x_2|, \dotsc, |x_n|,|x_{n+1}|, \ldots)\] and the corresponding space \(\ell^\infty\) of all bounded sequences. It turns out that \[\|x\|_\infty = \lim_{p \to \infty} \|x\|_p\] if the right-hand side is finite, or the left-hand side is infinite. Thus, we will consider \(\ell^p\) spaces for \(1 \leq p \leq \infty.\)

The \(p\)-norm thus defined on \(\ell^p\) is indeed a norm, and \(\ell^p\) together with this norm is a Banach space.

In complete analogy to the preceding definition one can define the space \(\ell^p(I)\) over a general index set \(I\) (and \(1 \leq p < \infty\)) as \[\ell^p(I) = \left\{(x_i)_{i\in I} \in \mathbb{K}^I : \sum_{i \in I} |x_i|^p < +\infty\right\},\] where convergence on the right requires that only countably many summands are nonzero (see also Absolute convergence over sets). With the norm \[\|x\|_p = \left(\sum_{i\in I} |x_i|^p\right)^{1/p}\] the space \(\ell^p(I)\) becomes a Banach space. In the case where \(I\) is finite with \(n\) elements, this construction yields \(\Reals^n\) with the \(p\)-norm defined above. If \(I\) is countably infinite, this is exactly the sequence space \(\ell^p\) defined above. For uncountable sets \(I\) this is a non-separable Banach space which can be seen as the locally convex direct limit of \(\ell^p\)-sequence spaces.

For \(p = 2,\) the \(\|\,\cdot\,\|_2\)-norm is even induced by a canonical inner product \(\langle \,\cdot,\,\cdot\rangle,\) called the Euclidean inner product, which means that \(\|\mathbf{x}\|_2 = \sqrt{\langle\mathbf{x}, \mathbf{x}\rangle}\) holds for all vectors \(\mathbf{x}.\) This inner product can be expressed in terms of the norm by using the polarization identity. On \(\ell^2,\) it can be defined by \[\langle \left(x_i\right)_{i}, \left(y_n\right)_{i} \rangle_{\ell^2} ~=~ \sum_i x_i \overline{y_i}.\] Now consider the case \(p = \infty.\) Define \[\ell^\infty(I)=\{x\in \mathbb K^I : \sup\operatorname{range}|x|<+\infty\},\] where for all \(x\) \[\|x\|_\infty\equiv\inf\{C \in \Reals_{\geq 0}:|x_i| \leq C\text{ for all } i \in I\} = \begin{cases}\sup\operatorname{range}|x|&\text{if } X\neq\varnothing,\\0&\text{if } X=\varnothing.\end{cases}\]

The index set \(I\) can be turned into a measure space by giving it the discrete σ-algebra and the counting measure. Then the space \(\ell^p(I)\) is just a special case of the more general \(L^p\)-space (defined below).

An \(L^p\) space may be defined as a space of measurable functions for which the \(p\)-th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let \((S, \Sigma, \mu)\) be a measure space and \(1 \leq p \leq \infty.\) When \(p \neq \infty\), consider the set \(\mathcal{L}^p(S,\, \mu)\) of all measurable functions \(f\) from \(S\) to \(\Complex\) or \(\Reals\) whose absolute value raised to the \(p\)-th power has a finite integral, or in symbols: \[\|f\|_p ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left(\int_S |f|^p\;\mathrm{d}\mu\right)^{1/p} < \infty.\]

To define the set for \(p = \infty,\) recall that two functions \(f\) and \(g\) defined on \(S\) are said to be equal almost everywhere, written \(f = g\) a.e., if the set \(\{s \in S : f(s) \neq g(s)\}\) is measurable and has measure zero. Similarly, a measurable function \(f\) (and its absolute value) is bounded (or dominated) almost everywhere by a real number \(C,\) written \(|f| \leq C\) a.e., if the (necessarily) measurable set \(\{s \in S : |f(s)| > C\}\) has measure zero. The space \(\mathcal{L}^\infty(S,\mu)\) is the set of all measurable functions \(f\) that are bounded almost everywhere (by some real \(C\)) and \(\|f\|_\infty\) is defined as the infimum of these bounds: \[\|f\|_\infty ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \inf \{C \in \Reals_{\geq 0} : |f(s)| \leq C \text{ for almost every } s\}.\] When \(\mu(S) \neq 0\) then this is the same as the essential supremum of the absolute value of \(f\): \[\|f\|_\infty ~=~ \begin{cases}\operatorname{esssup}|f| & \text{if } \mu(S) > 0,\\ 0 & \text{if } \mu(S) = 0.\end{cases}\]

For example, if \(f\) is a measurable function that is equal to \(0\) almost everywhere then \(\|f\|_p = 0\) for every \(p\) and thus \(f \in \mathcal{L}^p(S,\, \mu)\) for all \(p.\)

For every positive \(p,\) the value under \(\|\,\cdot\,\|_p\) of a measurable function \(f\) and its absolute value \(|f| : S \to [0, \infty]\) are always the same (that is, \(\|f\|_p = \||f|\|_p\) for all \(p\)) and so a measurable function belongs to \(\mathcal{L}^p(S,\, \mu)\) if and only if its absolute value does. Because of this, many formulas involving \(p\)-norms are stated only for non-negative real-valued functions. Consider for example the identity \(\|f\|_p^r = \|f^r\|_{p/r},\) which holds whenever \(f \geq 0\) is measurable, \(r > 0\) is real, and 0 < p \leq \infty\) (here \(\infty / r \;\stackrel{\scriptscriptstyle\text{def}}{=}\; \infty\) when \(p = \infty\)). The non-negativity requirement \(f \geq 0\) can be removed by substituting \(|f|\) in for \(f,\) which gives \(\|\,|f|\,\|_p^r = \|\,|f|^r\,\|_{p/r}.\) Note in particular that when \(p = r\) is finite then the formula \(\|f\|_p^p = \||f|^p\|_1\) relates the \(p\)-norm to the \(1\)-norm.

Seminormed space of \(p\)-th power integrable functions

Each set of functions \(\mathcal{L}^p(S,\, \mu)\) forms a vector space when addition and scalar multiplication are defined pointwise. That the sum of two \(p\)-th power integrable functions \(f\) and \(g\) is again \(p\)-th power integrable follows from \(\|f + g\|_p^p \leq 2^{p-1} \left(\|f\|_p^p + \|g\|_p^p\right),\) although it is also a consequence of Minkowski's inequality \[\|f + g\|_p \leq \|f\|_p + \|g\|_p\] which establishes that \(\|\cdot\|_p\) satisfies the triangle inequality for \(1 \leq p \leq \infty\) (the triangle inequality does not hold for 0 < p < 1\)). That \(\mathcal{L}^p(S,\, \mu)\) is closed under scalar multiplication is due to \(\|\cdot\|_p\) being absolutely homogeneous, which means that \(\|s f\|_p = |s| \|f\|_p\) for every scalar \(s\) and every function \(f.\)

Condensed: the full section is in Wikipedia.

For \(1 \leq p \leq \infty\) the \(\ell^p\) spaces are a special case of \(L^p\) spaces; when \(S\) are the natural numbers \(\mathbb{N}\) and \(\mu\) is the counting measure. More generally, if one considers any set \(S\) with the counting measure, the resulting \(L^p\) space is denoted \(\ell^p(S).\) For example, \(\ell^p(\mathbb{Z})\) is the space of all sequences indexed by the integers, and when defining the \(p\)-norm on such a space, one sums over all the integers. The space \(\ell^p(n),\) where \(n\) is the set with \(n\) elements, is \(\Reals^n\) with its \(p\)-norm as defined above.

Similar to \(\ell^2\) spaces, \(L^2\) is the only Hilbert space among \(L^p\) spaces. In the complex case, the inner product on \(L^2\) is defined by \[\langle f, g \rangle = \int_S f(x) \overline{g(x)} \, \mathrm{d}\mu(x).\] Functions in \(L^2\) 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).

As any Hilbert space, every space \(L^2\) is linearly isometric to a suitable \(\ell^2(I),\) where the cardinality of the set \(I\) is the cardinality of an arbitrary basis for this particular \(L^2.\)

If we use complex-valued functions, the space \(L^\infty\) 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 \(L^\infty\) defines a bounded operator on any \(L^p\) space by multiplication.

If 0 < p < 1,\) then \(L^p(\mu)\) can be defined as above, that is: \[N_p(f) = \int_S |f|^p\, d\mu < \infty.\] In this case, however, the \(p\)-norm \(\|f\|_p = N_p(f)^{1/p}\) does not satisfy the triangle inequality and defines only a quasi-norm. The inequality \((a + b)^p \leq a^p + b^p,\) valid for \(a, b \geq 0,\) implies that \[N_p(f + g) \leq N_p(f) + N_p(g)\] and so the function \[d_p(f ,g) = N_p(f - g) = \|f - g\|_p^p\] is a metric on \(L^p(\mu).\) The resulting metric space is complete, but not locally convex.

In this setting \(L^p\) satisfies a reverse Minkowski inequality, that is for \(u, v \in L^p\) \[\Big\||u| + |v|\Big\|_p \geq \|u\|_p + \|v\|_p\]

This result may be used to prove Clarkson's inequalities, which are in turn used to establish the uniform convexity of the spaces \(L^p\) for 1 < p < \infty\) (Adams & Fournier 2003).

The space \(L^p\) for 0 < p < 1\) is an F-space: it admits a complete translation-invariant metric with respect to which the vector space operations are continuous. It is the prototypical example of an F-space that, for most reasonable measure spaces, is not locally convex: in \(\ell^p\) or \(L^p([0, 1]),\) every open convex set containing the \(0\) function is unbounded for the \(p\)-quasi-norm; therefore, the \(0\) vector does not possess a fundamental system of convex neighborhoods. Specifically, this is true if the measure space \(S\) contains an infinite family of disjoint measurable sets of finite positive measure.

The only nonempty convex open set in \(L^p([0, 1])\) is the entire space. Consequently, there are no nonzero continuous linear functionals on \(L^p([0, 1]);\) the continuous dual space is the zero space. In the case of the counting measure on the natural numbers (i.e. \(L^p(\mu) = \ell^p\)), the bounded linear functionals on \(\ell^p\) are exactly those that are bounded on \(\ell^1\), i.e., those given by sequences in \(\ell^\infty.\) Although \(\ell^p\) does contain non-trivial convex open sets, it fails to have enough of them to give a base for the topology.

Having no linear functionals is highly undesirable for the purposes of doing analysis. In case of the Lebesgue measure on \(\Reals^n,\) rather than work with \(L^p\) for 0 < p < 1,\) it is common to work with the Hardy space H whenever possible, as this has quite a few linear functionals: enough to distinguish points from one another. However, the Hahn-Banach theorem still fails in H for \(p < 1\) (Duren 1970, §7.5).

For a finite measure space \((X,\mu)\), \(L^0(\mu)\) is defined as the space of all \(\mu\)-measurable functions, with functions identified if they are equal almost everywhere. The F-norm is defined by \[\|f\|_0 = \int_X \frac{|f(x)|}{1+|f(x)|}d\mu(x).\] This is not a true norm. As with 0

The space \(\ell_0\) sometimes denotes the space of all sequences, with the F-norm \[\|f\|_0 = \sum_n 2^{-n}\frac{|f_n|}{1+|f_n|}.\] This is a special case of \(L^0\) with a weighted measure on the positive integers.

Suppose \(p, q, r \in [1, \infty]\) satisfy \(\tfrac{1}{p} + \tfrac{1}{q} = \tfrac{1}{r}\). If \(f \in L^p(S, \mu)\) and \(g \in L^q(S, \mu)\) then \(f g \in L^r(S, \mu)\) and \[\|f g\|_r ~\leq~ \|f\|_p \, \|g\|_q.\]

This inequality, called Hölder's inequality, is in some sense optimal since if \(r = 1\) and \(f\) is a measurable function such that \[\sup_{\|g\|_q \leq 1} \, \int_S |f g| \, \mathrm{d} \mu ~<~ \infty\] where the supremum is taken over the closed unit ball of \(L^q(S, \mu),\) then \(f \in L^p(S, \mu)\) and \[\|f\|_p ~=~ \sup_{\|g\|_q \leq 1} \, \int_S f g \, \mathrm{d} \mu.\]

Minkowski inequality, which states that \(\|\cdot\|_p\) satisfies the triangle inequality, can be generalized: If the measurable function \(F : M \times N \to \Reals\) is non-negative (where \((M, \mu)\) and \((N, \nu)\) are measure spaces) then for all \(1 \leq p \leq q \leq \infty,\) \[\left\|\left\|F(\,\cdot, n)\right\|_{L^p(M, \mu)}\right\|_{L^q(N, \nu)} ~\leq~ \left\|\left\|F(m, \cdot)\right\|_{L^q(N, \nu)}\right\|_{L^p(M, \mu)} \ .\]

If \(1 \leq p < \infty\) then every non-negative \(f \in L^p(\mu)\) has an atomic decomposition, meaning that there exist a sequence \((r_n)_{n \in \Z}\) of non-negative real numbers and a sequence of non-negative functions \((f_n)_{n \in \Z},\) called the atoms, whose supports \(\left(\operatorname{supp} f_n\right)_{n \in \Z}\) are pairwise disjoint sets of measure \(\mu\left(\operatorname{supp} f_n\right) \leq 2^{n+1},\) such that \[f ~=~ \sum_{n \in \Z} r_n \, f_n \, ,\] and for every integer \(n \in \Z,\) \[\|f_n\|_\infty ~\leq~ 2^{-\tfrac{n}{p}} \, ,\] and \[\tfrac{1}{2} \|f\|_p^p ~\leq~ \sum_{n \in \Z} r_n^p ~\leq~ 2 \|f\|^p_p \, ,\] and where moreover, the sequence of functions \((r_n f_n)_{n \in\Z}\) depends only on \(f\) (it is independent of \(p\)). These inequalities guarantee that \(\|f_n\|_p^p \leq 2\) for all integers \(n\) while the supports of \((f_n)_{n \in \Z}\) being pairwise disjoint implies \[\|f\|_p^p ~=~ \sum_{n \in \Z} r_n^p \, \|f_n\|^p_p \, .\]

An atomic decomposition can be explicitly given by first defining for every integer \(n \in \Z,\) \[t_n = \inf \{t \in \Reals : \mu(f > t) < 2^n\}\] and then letting \[r_n ~=~ 2^{n/p} \, t_n ~ \text{ and } \quad f_n ~=~ \frac{f}{r_n} \, \mathbf{1}_{( t_{n+1} < f \leq t_n )}\] where \(\mu(f > t) = \mu(\{s : f(s) > t\})\) denotes the measure of the set \((f > t) := \{s \in S : f(s) > t\}\) and \mathbf{1}_{(t_{n+1} < f \leq t_n)}\) denotes the indicator function of the set (t_{n+1} < f \leq t_n) := \{s \in S : t_{n+1} < f(s) \leq t_n\}.\) The sequence \((t_n)_{n \in \Z}\) is decreasing and converges to \(0\) as \(n \to \infty.\) Consequently, if \(t_n = 0\) then \(t_{n+1} = 0\) and (t_{n+1} < f \leq t_n) = \varnothing\) so that f_n = \frac{1}{r_n} \, f \,\mathbf{1}_{(t_{n+1} < f \leq t_n)}\) is identically equal to \(0\) (in particular, the division \(\tfrac{1}{r_n}\) by \(r_n = 0\) causes no issues).

The complementary cumulative distribution function \(t \in \Reals \mapsto \mu(|f| > t)\) of \(|f| = f\) that was used to define the \(t_n\) also appears in the definition of the weak \(L^p\)-norm (given below) and can be used to express the \(p\)-norm \(\|\cdot\|_p\) (for \(1 \leq p < \infty\)) of \(f \in L^p(S, \mu)\) as the integral \[\|f\|_p^p ~=~ p \, \int_0^\infty t^{p-1} \mu(|f| > t) \, \mathrm{d} t \, ,\] where the integration is with respect to the usual Lebesgue measure on \((0, \infty).\)

The dual space of \(L^p(\mu)\) for 1 < p < \infty\) has a natural isomorphism with \(L^q(\mu),\) where \(q\) is such that \(\tfrac{1}{p} + \tfrac{1}{q} = 1\). This isomorphism associates \(g \in L^q(\mu)\) with the functional \(\kappa_p(g) \in L^p(\mu)^*\) defined by \[f \mapsto \kappa_p(g)(f) = \int f g \, \mathrm{d}\mu\] for every \(f \in L^p(\mu).\)

\(\kappa_p : L^q(\mu) \to L^p(\mu)^*\) is a well defined continuous linear mapping which is an isometry by the extremal case of Hölder's inequality. If \((S,\Sigma,\mu)\) is a \(\sigma\)-finite measure space one can use the Radon-Nikodym theorem to show that any \(G \in L^p(\mu)^*\) can be expressed this way, i.e., \(\kappa_p\) is an isometric isomorphism of Banach spaces. Hence, it is usual to say simply that \(L^q(\mu)\) is the continuous dual space of \(L^p(\mu).\)

For 1 < p < \infty,\) the space \(L^p(\mu)\) is reflexive. Let \(\kappa_p\) be as above and let \(\kappa_q : L^p(\mu) \to L^q(\mu)^*\) be the corresponding linear isometry. Consider the map from \(L^p(\mu)\) to \(L^p(\mu)^{**},\) obtained by composing \(\kappa_q\) with the transpose (or adjoint) of the inverse of \(\kappa_p:\)

\[j_p : L^p(\mu) \mathrel{\overset{\kappa_q}{\longrightarrow}} L^q(\mu)^* \mathrel{\overset{\left(\kappa_p^{-1}\right)^*}{\longrightarrow}} L^p(\mu)^{**}\]

This map coincides with the canonical embedding \(J\) of \(L^p(\mu)\) into its bidual. Moreover, the map \(j_p\) is onto, as composition of two onto isometries, and this proves reflexivity.

If the measure \(\mu\) on \(S\) is sigma-finite, then the dual of \(L^1(\mu)\) is isometrically isomorphic to \(L^\infty(\mu)\) (more precisely, the map \(\kappa_1\) corresponding to \(p = 1\) is an isometry from \(L^\infty(\mu)\) onto \(L^1(\mu)^*.\)

The dual of \(L^\infty(\mu)\) is subtler. Elements of \(L^\infty(\mu)^*\) can be identified with bounded signed finitely additive measures on \(S\) that are absolutely continuous with respect to \(\mu.\) See ba space for more details. If we assume the axiom of choice, this space is much bigger than \(L^1(\mu)\) except in some trivial cases. However, Saharon Shelah proved that there are relatively consistent extensions of Zermelo-Fraenkel set theory (ZF + DC + "Every subset of the real numbers has the Baire property") in which the dual of \(\ell^\infty\) is \(\ell^1.\)

Colloquially, if 1 \leq p < q \leq \infty,\) then \(L^p(S, \mu)\) contains functions that are more locally singular, while elements of \(L^q(S, \mu)\) can be more spread out. Consider the Lebesgue measure on the half line \((0, \infty).\) A continuous function in \(L^1\) might blow up near \(0\) but must decay sufficiently fast toward infinity. On the other hand, continuous functions in \(L^\infty\) need not decay at all but no blow-up is allowed. More formally:

1. If 0
2. If 0

Neither condition holds for the Lebesgue measure on the real line while both conditions holds for the counting measure on any finite set. As a consequence of the closed graph theorem, the embedding is continuous, i.e., the identity operator is a bounded linear map from \(L^q\) to \(L^p\) in the first case and \(L^p\) to \(L^q\) in the second. Indeed, if the domain \(S\) has finite measure, one can make the following explicit calculation using Hölder's inequality \[\ \|\mathbf{1}f^p\|_1 \leq \|\mathbf{1}\|_{q/(q-p)} \|f^p\|_{q/p}\] leading to \[\ \|f\|_p \leq \mu(S)^{1/p - 1/q} \|f\|_q .\]

The constant appearing in the above inequality is optimal, in the sense that the operator norm of the identity \(I : L^q(S, \mu) \to L^p(S, \mu)\) is precisely \[\|I\|_{q,p} = \mu(S)^{1/p - 1/q}\] the case of equality being achieved exactly when \(f = 1\) \(\mu\)-almost-everywhere.

Let \(1 \leq p < \infty\) and \((S, \Sigma, \mu)\) be a measure space and consider an integrable simple function \(f\) on \(S\) given by \[f = \sum_{j=1}^n a_j \mathbf{1}_{A_j},\] where \(a_j\) are scalars, \(A_j \in \Sigma\) has finite measure and \({\mathbf 1}_{A_j}\) is the indicator function of the set \(A_j,\) for \(j = 1, \dots, n.\) By construction of the integral, the vector space of integrable simple functions is dense in \(L^p(S, \Sigma, \mu).\)

More can be said when \(S\) is a normal topological space and \(\Sigma\) its Borel 𝜎, algebra.

Suppose \(V \subseteq S\) is an open set with \(\mu(V) < \infty.\) Then for every Borel set \(A \in \Sigma\) contained in \(V\) there exist a closed set \(F\) and an open set \(U\) such that \[F \subseteq A \subseteq U \subseteq V \quad \text{and} \quad \mu(U \setminus F)= \mu(U) - \mu(F) < \varepsilon,\] for every \(\varepsilon > 0\). Subsequently, there exists a Urysohn function \(0 \leq \varphi \leq 1\) on \(S\) that is \(1\) on \(F\) and \(0\) on \(S \setminus U,\) with \[\int_S |\mathbf{1}_A - \varphi| \, \mathrm{d}\mu < \varepsilon \, .\]

If \(S\) can be covered by an increasing sequence \((V_n)\) of open sets that have finite measure, then the space of \(p\)–integrable continuous functions is dense in \(L^p(S, \Sigma, \mu).\) More precisely, one can use bounded continuous functions that vanish outside one of the open sets \(V_n.\)

This applies in particular when \(S = \Reals^d\) and when \(\mu\) is the Lebesgue measure. For example, the space of continuous and compactly supported functions as well as the space of integrable step functions are dense in \(L^p(\Reals^d)\).