$L^p$ Spaces

Published

Last modified: 2026-10-31 15:22:31 (PDT)

0.1 $L^p$ spaces

Definition 1 ($L^p$ norm ($p$-norm of a function))

Definition 1 ($L^p$ norm ($p$-norm of a function)) Let $(S, \mathcal{S}, \mu)$ be a measure space. For any real number $p \ge 1$, the $L^p$ norm of a measurable function $f : S \to \mathbb{R}$ is:

\[ \mathopen{}\left\lVert f\right\rVert\mathclose{}_p \stackrel{\text{def}}{=}\mathopen{}\left(\int_S \mathopen{}\left|f\right|\mathclose{}^p \, d\mu\right)\mathclose{}^{1/p}. \]

Example 1 (The $L^1$ and $L^2$ norms of $x \mapsto x$ on the unit interval)

Example 1 (The $L^1$ and $L^2$ norms of $x \mapsto x$ on the unit interval) Let $S = [0, 1]$ with the Lebesgue measure $\mu$, and let $f(x) = x$. By Definition 1,

\[ \begin{aligned} \mathopen{}\left\lVert f\right\rVert\mathclose{}_1 &= \int_0^1 x \, dx \\ &= \frac{1}{2}, \end{aligned} \]

and

\[ \begin{aligned} \mathopen{}\left\lVert f\right\rVert\mathclose{}_2 &= \mathopen{}\left(\int_0^1 x^2 \, dx\right)\mathclose{}^{1/2} \\ &= \mathopen{}\left(\frac{1}{3}\right)\mathclose{}^{1/2}. \end{aligned} \]

Show code

lpna_x_pf <- seq(0, 1, length.out = 201)
lpna_p1_pf <- tibble::tibble(
  x = lpna_x_pf, y = lpna_x_pf, panel = "p = 1: area = 1/2"
)
lpna_sq_pf <- lpna_x_pf^2
lpna_p2_pf <- tibble::tibble(
  x = lpna_x_pf, y = lpna_sq_pf, panel = "p = 2: area = 1/3"
)
lpna_curves_pf <- dplyr::bind_rows(lpna_p1_pf, lpna_p2_pf)
lpna_aes_pf <- ggplot2::aes(x, y)
ggplot2::ggplot(lpna_curves_pf, lpna_aes_pf) +
  ggplot2::geom_area(fill = "#0072B2", alpha = 0.3) +
  ggplot2::geom_line(colour = "#0072B2") +
  ggplot2::facet_wrap(ggplot2::vars(panel)) +
  ggplot2::labs(x = "x", y = "|f(x)|^p")

Show code

import matplotlib.pyplot as plt
import numpy as np

lpna_x_pf = np.linspace(0, 1, 201)
fig, axes = plt.subplots(1, 2, figsize=(7, 3.5), sharey=True,
                         layout="constrained")
for ax, p, title in zip(axes, [1, 2], ["p = 1: area = 1/2", "p = 2: area = 1/3"]):
    ax.fill_between(lpna_x_pf, lpna_x_pf**p, color="#0072B2", alpha=0.3)
    ax.plot(lpna_x_pf, lpna_x_pf**p, color="#0072B2")
    ax.set_title(title)
    ax.set_xlabel("x")
axes[0].set_ylabel("|f(x)|^p")
plt.show()

Two side-by-side plots of a curve over x from 0 to 1, each with the area under the curve shaded blue. The left plot, titled p equals 1, shows the straight line from 0 at x of 0 to 1 at x of 1, with a shaded triangle of area one half. The right plot, titled p equals 2, shows the parabola from 0 to 1, with the shaded region under it of area one third. — Figure 1: The area under $\mathopen{}\left|f\right|\mathclose{}^p$ for $f(x) = x$ on $[0, 1]$ in Example 1. The area is $\mathopen{}\left\lVert f\right\rVert\mathclose{}_p^p$: $1/2$ for $p = 1$ and $1/3$ for $p = 2$, so $\mathopen{}\left\lVert f\right\rVert\mathclose{}_2 = \sqrt{1/3}$.

Definition 2 ($\mathcal{L}^p$ space ($\mathcal{L}^p(S, \mathcal{S}, \mu)$))

Definition 2 ($\mathcal{L}^p$ space ($\mathcal{L}^p(S, \mathcal{S}, \mu)$)) Let $(S, \mathcal{S}, \mu)$ be a measure space. Let $p \ge 1$ be a real number. The $\mathcal{L}^p$ space $\mathcal{L}^p(S, \mathcal{S}, \mu)$ is the set of all measurable functions $f : S \to \mathbb{R}$ whose $L^p$ norm is less than $\infty$: $\mathopen{}\left\lVert f\right\rVert\mathclose{}_p < \infty$.

Example 2 (An $L^1$ function not in $L^2$)

Example 2 (An $L^1$ function not in $L^2$) Consider the interval $(0, 1)$ with the Borel $\sigma$-algebra (Definition 3 in Sigma-Algebras) and Lebesgue measure $\mu$ (Definition 6 in Measures and Probability Measures). Define $f : (0, 1) \to \mathbb{R}$ by $f(x) = x^{-1/2}$.

For $p = 1$: \[ \begin{aligned} \mathopen{}\left\lVert f\right\rVert\mathclose{}_1 &= \int_0^1 x^{-1/2} \, dx \\ &= \mathopen{}\left[2 x^{1/2}\right]\mathclose{}_0^1 \\ &= 2 \\ &< \infty, \end{aligned} \] so $f \in \mathcal{L}^1((0, 1), \mu)$.
For $p = 2$: \[ \begin{aligned} \mathopen{}\left|f(x)\right|\mathclose{}^2 &= \mathopen{}\left(x^{-1/2}\right)\mathclose{}^2 \\ &= \frac{1}{x}. \end{aligned} \] The integral is: \[ \begin{aligned} \int_0^1 \frac{1}{x} \, dx &= \lim_{\varepsilon\to 0^+} \mathopen{}\left[\ln x\right]\mathclose{}_\varepsilon^1 \\ &= \lim_{\varepsilon\to 0^+} (-\ln \varepsilon) \\ &= \infty. \end{aligned} \] So $\mathopen{}\left\lVert f\right\rVert\mathclose{}_2 = \infty$, so $f \notin \mathcal{L}^2((0, 1), \mu)$.

This shows that $\mathcal{L}^2 \subset \mathcal{L}^1$ on sets of finite measure: a larger $p$ imposes stricter requirements near singularities.

Show code

lpss_x_pf <- seq(0.01, 1, length.out = 400)
lpss_root_pf <- lpss_x_pf^(-1 / 2)
lpss_p1_pf <- tibble::tibble(
  x = lpss_x_pf, y = lpss_root_pf, panel = "p = 1: area = 2"
)
lpss_inv_pf <- 1 / lpss_x_pf
lpss_p2_pf <- tibble::tibble(
  x = lpss_x_pf, y = lpss_inv_pf, panel = "p = 2: area = infinity"
)
lpss_curves_pf <- dplyr::bind_rows(lpss_p1_pf, lpss_p2_pf)
lpss_aes_pf <- ggplot2::aes(x, y, fill = panel, colour = panel)
lpss_colours_pf <- c("#0072B2", "#D55E00")
lpss_ylim_pf <- c(0, 10)
ggplot2::ggplot(lpss_curves_pf, lpss_aes_pf) +
  ggplot2::geom_area(alpha = 0.3, colour = NA) +
  ggplot2::geom_line() +
  ggplot2::facet_wrap(ggplot2::vars(panel)) +
  ggplot2::scale_fill_manual(values = lpss_colours_pf) +
  ggplot2::scale_colour_manual(values = lpss_colours_pf) +
  ggplot2::coord_cartesian(ylim = lpss_ylim_pf) +
  ggplot2::guides(fill = "none", colour = "none") +
  ggplot2::labs(x = "x", y = "|f(x)|^p")

Show code

import matplotlib.pyplot as plt
import numpy as np

lpss_x_pf = np.linspace(0.01, 1, 400)
lpss_curves_pf = [
    ("p = 1: area = 2", lpss_x_pf ** (-1 / 2), "#0072B2"),
    ("p = 2: area = infinity", 1 / lpss_x_pf, "#D55E00"),
]
fig, axes = plt.subplots(1, 2, figsize=(7, 3.5), sharey=True,
                         layout="constrained")
for ax, (title, y, colour) in zip(axes, lpss_curves_pf):
    ax.fill_between(lpss_x_pf, y, color=colour, alpha=0.3)
    ax.plot(lpss_x_pf, y, color=colour)
    ax.set_title(title)
    ax.set_xlabel("x")
    ax.set_ylim(0, 10)
#> <matplotlib.collections.FillBetweenPolyCollection object at 0x7f5dc77c2490>
#> [<matplotlib.lines.Line2D object at 0x7f5dc77c25d0>]
#> Text(0.5, 1.0, 'p = 1: area = 2')
#> Text(0.5, 0, 'x')
#> (0.0, 10.0)
#> <matplotlib.collections.FillBetweenPolyCollection object at 0x7f5dc77c2710>
#> [<matplotlib.lines.Line2D object at 0x7f5dc77c2850>]
#> Text(0.5, 1.0, 'p = 2: area = infinity')
#> Text(0.5, 0, 'x')
#> (0.0, 10.0)
axes[0].set_ylabel("|f(x)|^p")
plt.show()

Two side-by-side plots over x from 0 to 1, with the vertical axis cut off at 10. The left plot, titled p equals 1, shows the curve one over the square root of x falling steeply from 10 at x of 0.01 to 1 at x of 1, with the area under it shaded blue; the area is 2. The right plot, titled p equals 2, shows the curve one over x, which rises past 10 just left of x of 0.1 and keeps climbing toward x of 0. The area under it is shaded vermilion and runs off the top of the plot, because its area is infinite. — Figure 2: The area under $\mathopen{}\left|f\right|\mathclose{}^p$ for $f(x) = x^{-1/2}$ on $(0, 1)$ in Example 2. The curve for $p = 1$ has area $2$, so $f \in \mathcal{L}^1$. The curve for $p = 2$ keeps climbing toward $x = 0$ and its area is $\infty$, so $f \notin \mathcal{L}^2$. The plots cut the curves off at $10$.

Definition 3 (Equal almost everywhere ($f = g$ $\mu$-a.e.))

Definition 3 (Equal almost everywhere ($f = g$ $\mu$-a.e.)) Let $(S, \mathcal{S}, \mu)$ be a measure space. Let $f, g : S \to \mathbb{R}$ be measurable. The functions $f$ and $g$ are equal almost everywhere, written $f = g$ $\mu$-a.e., if $f(s) = g(s)$ holds almost everywhere.

Example 3 (A function that differs from $0$ at one point)

Example 3 (A function that differs from $0$ at one point) Let $S = \mathbb{R}$ with the Lebesgue measure $\mu$. Let $f(s) = 1$ if $s = 0$, and $f(s) = 0$ otherwise. The set ${\mathopen{}\left\{s : f(s) \neq 0\right\}\mathclose{}} = {\mathopen{}\left\{0\right\}\mathclose{}}$ has measure zero, so $f = 0$ almost everywhere by Definition 3.

Definition 4 ($L^p$ space ($L^p(S, \mathcal{S}, \mu)$, $L^p(\mu)$))

Definition 4 ($L^p$ space ($L^p(S, \mathcal{S}, \mu)$, $L^p(\mu)$)) Let $(S, \mathcal{S}, \mu)$ be a measure space. Let $p \ge 1$ be a real number. For each $f$ in the $\mathcal{L}^p$ space, let $[f]$ be the set of all functions in $\mathcal{L}^p(S, \mathcal{S}, \mu)$ that are equal almost everywhere to $f$. The $L^p$ space $L^p(S, \mathcal{S}, \mu)$ (or simply $L^p(\mu)$) is the set of all such sets $[f]$, with the norm $\mathopen{}\left\lVert[f]\right\rVert\mathclose{}_p \stackrel{\text{def}}{=}\mathopen{}\left\lVert f\right\rVert\mathclose{}_p$ ($L^p$ norm).

Example 4 (The zero element of $L^p$)

Example 4 (The zero element of $L^p$) Let $f$ be the function in Example 3, which is $1$ at $s = 0$ and $0$ elsewhere. Then $f = 0$ almost everywhere, so $[f] = [0]$. Also $\mathopen{}\left\lVert f\right\rVert\mathclose{}_p = 0$ although $f \neq 0$, so $\mathopen{}\left\lVert f\right\rVert\mathclose{}_p = 0$ can hold for a nonzero element of $\mathcal{L}^p$; on $L^p(\mu)$, $\mathopen{}\left\lVert[f]\right\rVert\mathclose{}_p = 0$ only for $[f] = [0]$, as a norm requires.

Definition 5 (Essential supremum norm ($L^\infty$ norm))

Definition 5 (Essential supremum norm ($L^\infty$ norm)) Let $(S, \mathcal{S}, \mu)$ be a measure space. The essential supremum norm (or $L^\infty$ norm) of a measurable function $f : S \to \mathbb{R}$ is:

\[ \mathopen{}\left\lVert f\right\rVert\mathclose{}_\infty \stackrel{\text{def}}{=}\inf \mathopen{}\left\{ C \ge 0 : \mu(\mathopen{}\left\{s \in S : \mathopen{}\left|f(s)\right|\mathclose{} > C\right\}\mathclose{}) = 0 \right\}\mathclose{}. \]

Example 5 (The essential supremum norm ignores a null set)

Example 5 (The essential supremum norm ignores a null set) Let $S = [0, 1]$ with the Lebesgue measure $\mu$. Let $f(s) = 5$ if $s = 0$ and $f(s) = s$ otherwise. Then $\sup_s \mathopen{}\left|f(s)\right|\mathclose{} = 5$. For $C \ge 1$, the set ${\mathopen{}\left\{s : \mathopen{}\left|f(s)\right|\mathclose{} > C\right\}\mathclose{}}$ is empty or ${\mathopen{}\left\{0\right\}\mathclose{}}$, so it has measure zero, and for $C < 1$ it has positive measure. By Definition 5, $\mathopen{}\left\lVert f\right\rVert\mathclose{}_\infty = 1$.

Show code

esn_x_pf <- seq(0, 1, length.out = 101)
esn_line_pf <- tibble::tibble(x = esn_x_pf, y = esn_x_pf)
esn_open_pf <- tibble::tibble(x = 0, y = 0)
esn_spike_pf <- tibble::tibble(x = 0, y = 5)
esn_aes_pf <- ggplot2::aes(x, y)
ggplot2::ggplot(esn_line_pf, esn_aes_pf) +
  ggplot2::geom_line(colour = "#0072B2") +
  ggplot2::geom_point(data = esn_open_pf, shape = 1, size = 3) +
  ggplot2::geom_point(data = esn_spike_pf, colour = "#D55E00", size = 3) +
  ggplot2::geom_hline(yintercept = 1, linetype = "dashed") +
  ggplot2::labs(x = "s", y = "f(s)")

Show code

import matplotlib.pyplot as plt
import numpy as np

esn_x_pf = np.linspace(0, 1, 101)
fig, ax = plt.subplots(figsize=(5, 3.5), layout="constrained")
ax.plot(esn_x_pf, esn_x_pf, color="#0072B2")
ax.plot(0, 0, marker="o", markerfacecolor="white", color="black")
ax.plot(0, 5, marker="o", color="#D55E00")
ax.axhline(1, linestyle="dashed", color="black")
ax.set_xlabel("s")
ax.set_ylabel("f(s)")
plt.show()

Plot of a function on x from 0 to 1. A solid blue diagonal line rises from an open circle at the origin to the point 1 at x of 1. A single vermilion filled point sits far above, at the value 5 at x of 0. A dashed horizontal line at height 1 marks the essential supremum. — Figure 3: The function in Example 5: $f(s) = s$ except $f(0) = 5$. The supremum is $5$, reached only at the single point $s = 0$, which has measure zero, so the essential supremum norm is the dashed height $1$.

Definition 6 ($\mathcal{L}^\infty$ space ($\mathcal{L}^\infty(S, \mathcal{S}, \mu)$))

Definition 6 ($\mathcal{L}^\infty$ space ($\mathcal{L}^\infty(S, \mathcal{S}, \mu)$)) Let $(S, \mathcal{S}, \mu)$ be a measure space. The $\mathcal{L}^\infty$ space $\mathcal{L}^\infty(S, \mathcal{S}, \mu)$ is the set of all measurable functions $f : S \to \mathbb{R}$ whose essential supremum norm is less than $\infty$: $\mathopen{}\left\lVert f\right\rVert\mathclose{}_\infty < \infty$.

Example 6 (Bounded and unbounded functions)

Example 6 (Bounded and unbounded functions) Let $S = (0, 1)$ with the Lebesgue measure $\mu$. The function $f(s) = s$ has $\mathopen{}\left\lVert f\right\rVert\mathclose{}_\infty = 1 < \infty$, so $f \in \mathcal{L}^\infty(S, \mathcal{S}, \mu)$ by Definition 6. The function $g(s) = 1/s$ exceeds every $C$ on the set $(0, 1/C)$ of positive measure, so $\mathopen{}\left\lVert g\right\rVert\mathclose{}_\infty = \infty$ and $g \notin \mathcal{L}^\infty(S, \mathcal{S}, \mu)$.

Definition 7 ($L^\infty$ space ($L^\infty(S, \mathcal{S}, \mu)$, $L^\infty(\mu)$))

Definition 7 ($L^\infty$ space ($L^\infty(S, \mathcal{S}, \mu)$, $L^\infty(\mu)$)) Let $(S, \mathcal{S}, \mu)$ be a measure space. For each $f$ in the $\mathcal{L}^\infty$ space, let $[f]$ be the set of all functions in $\mathcal{L}^\infty(S, \mathcal{S}, \mu)$ that are equal almost everywhere to $f$. The $L^\infty$ space $L^\infty(S, \mathcal{S}, \mu)$ (or simply $L^\infty(\mu)$) is the set of all such sets $[f]$, with the norm $\mathopen{}\left\lVert[f]\right\rVert\mathclose{}_\infty \stackrel{\text{def}}{=}\mathopen{}\left\lVert f\right\rVert\mathclose{}_\infty$ (essential supremum norm).

Example 7 (Bounded functions equal almost everywhere)

Example 7 (Bounded functions equal almost everywhere) Let $f$ be the function in Example 5 and $h(s) = s$ on $[0, 1]$. Then $f = h$ almost everywhere (Definition 3), so $[f] = [h]$ in $L^\infty(\mu)$, and $\mathopen{}\left\lVert[f]\right\rVert\mathclose{}_\infty = \mathopen{}\left\lVert h\right\rVert\mathclose{}_\infty = 1$.

Theorem 1 (Holder’s inequality)

Theorem 1 (Holder’s inequality) Let $(S, \mathcal{S}, \mu)$ be a measure space, and let $p, q \in [1, \infty]$ be conjugate exponents satisfying $\frac{1}{p} + \frac{1}{q} = 1$ (with the convention that $q = \infty$ when $p = 1$, and $q = 1$ when $p = \infty$). If $f \in L^p(\mu)$ and $g \in L^q(\mu)$ (Definition 4), then their product $f g$ is absolutely integrable ($f g \in L^1(\mu)$) and satisfies:

\[ \mathopen{}\left\lVert f g\right\rVert\mathclose{}_1 = \int_S \mathopen{}\left|f g\right|\mathclose{} \, d\mu\le \mathopen{}\left\lVert f\right\rVert\mathclose{}_p\,\mathopen{}\left\lVert g\right\rVert\mathclose{}_q. \]

When

\[ \begin{aligned} p &= q \\ &= 2, \end{aligned} \]

this inequality is the Cauchy-Schwarz inequality for integrals.

Theorem 2 (Minkowski’s inequality ($L^p$ triangle inequality))

Theorem 2 (Minkowski’s inequality ($L^p$ triangle inequality)) Let $(S, \mathcal{S}, \mu)$ be a measure space and let $1 \le p \le \infty$. For any functions $f, g \in L^p(\mu)$ (Definition 4), their sum $f + g$ is in $L^p(\mu)$ and satisfies:

\[ \mathopen{}\left\lVert f + g\right\rVert\mathclose{}_p \le \mathopen{}\left\lVert f\right\rVert\mathclose{}_p + \mathopen{}\left\lVert g\right\rVert\mathclose{}_p. \]

Together with absolute homogeneity $\mathopen{}\left\lVert c f\right\rVert\mathclose{}_p = \mathopen{}\left|c\right|\mathclose{}\,\mathopen{}\left\lVert f\right\rVert\mathclose{}_p$ and definiteness, Minkowski’s inequality establishes that $L^p(\mu)$ satisfies all axioms of a normed vector space.

Example 8 (Applying Holder’s inequality to finite sums and expectations)

Example 8 (Applying Holder’s inequality to finite sums and expectations)

Finite vectors (Cauchy-Schwarz and taxicab bounds): Equipping a finite set with the counting measure, Holder’s inequality for conjugate exponents $p, q$ gives: \[ \sum_{i=1}^n\mathopen{}\left|x_i y_i\right|\mathclose{} \le \mathopen{}\left(\sum_{i=1}^n\mathopen{}\left|x_i\right|\mathclose{}^p\right)\mathclose{}^{1/p} \mathopen{}\left(\sum_{i=1}^n\mathopen{}\left|y_i\right|\mathclose{}^q\right)\mathclose{}^{1/q}. \] Setting $p = 1$, $q = \infty$ gives $\sum_{i=1}^n\mathopen{}\left|x_i y_i\right|\mathclose{} \le \mathopen{}\left\lVert\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{x}}\vphantom{x}}\right\rVert\mathclose{}_1\,\mathopen{}\left\lVert\mathord{\smash[b]{\underset{\raise.3ex\hbox{$\scriptstyle\sim$}}{y}}\vphantom{y}}\right\rVert\mathclose{}_\infty$. Setting

\[ \begin{aligned} p &= q \\ &= 2 \end{aligned} \]

gives the standard vector Cauchy-Schwarz inequality.
Covariance bound for random variables: On a probability space with measure $\mathbb{P}$ (Definition 12 in Measures and Probability Measures), taking

\[ \begin{aligned} p &= q \\ &= 2 \end{aligned} \]

gives ${\mathbb{E}\mathopen{}\left[\mathopen{}\left|X Y\right|\mathclose{}\right]\mathclose{}} \le \sqrt{{\mathbb{E}\mathopen{}\left[X^2\right]\mathclose{}}}\,\sqrt{{\mathbb{E}\mathopen{}\left[Y^2\right]\mathclose{}}}$, guaranteeing that the covariance between any two square-integrable random variables is finite.

Remark 1 ($L^p$ spaces and norms in probability and machine learning)

Remark 1 ($L^p$ spaces and norms in probability and machine learning). In probability, statistics, and machine learning, $L^p$ spaces structure the geometry of random variables, estimators, and loss functions:

Random variables, moments, and inclusion of $L^p$ spaces: When $\mu= \mathbb{P}$ is a probability measure (Definition 12 in Measures and Probability Measures) with total measure ${\mathbb{P}\mathopen{}\left(S\right)\mathclose{}} = 1$, an $L^p$ function is a random variable $X$ with finite $p$-th moment: \[ \mathopen{}\left\lVert X\right\rVert\mathclose{}_p = \mathopen{}\left({\mathbb{E}\mathopen{}\left[\mathopen{}\left|X\right|\mathclose{}^p\right]\mathclose{}}\right)\mathclose{}^{1/p}. \] By Jensen’s inequality, if $1 \le p \le q \le \infty$, then $\mathopen{}\left\lVert X\right\rVert\mathclose{}_p \le \mathopen{}\left\lVert X\right\rVert\mathclose{}_q$, so $L^q(P) \subseteq L^p(P)$. For instance, every random variable with finite variance ($X \in L^2$) has finite expectation ($X \in L^1$).
The unique Hilbert space $L^2$ and conditional expectation: Among all $L^p$ spaces, only $L^2(\mu)$ is an inner product space, equipped with the inner product ${\mathopen{}\left\langle f, g\right\rangle\mathclose{}} \stackrel{\text{def}}{=}\int_S f g \, d\mu$ (all other $L^p$ spaces fail the parallelogram law). The completeness of $L^2$ makes it a Hilbert space. In regression, the conditional expectation ${\mathbb{E}\mathopen{}\left[Y \mid X\right]\mathclose{}}$ is the orthogonal projection of $Y \in L^2$ onto the closed subspace of $L^2$ functions measurable with respect to $X$, minimizing the population mean squared error (pMSE) $\mathopen{}\left\lVert Y - g(X)\right\rVert\mathclose{}_2^2$.
Loss functions, estimation, and robustness: The choice of $p$ in $L^p$ norms defines loss functions for optimization and regression:
- $L^2$ loss (population mean squared error (pMSE), or sample mean squared error (sMSE) on a sample): penalizes large errors quadratically, yielding the conditional mean; sensitive to outliers.
- $L^1$ loss (population mean absolute error (pMAE), or sample mean absolute error (sMAE) on a sample): penalizes errors linearly, yielding the conditional median; robust to heavy tails and outliers.
- $L^\infty$ loss (Chebyshev criterion): minimizes the worst-case maximum error across all instances.

Definitions on this page

Definition 1 ($L^p$ norm ($p$-norm of a function))
Definition 2 ($\mathcal{L}^p$ space ($\mathcal{L}^p(S, \mathcal{S}, \mu)$))
Definition 3 (Equal almost everywhere ($f = g$ $\mu$-a.e.))
Definition 4 ($L^p$ space ($L^p(S, \mathcal{S}, \mu)$, $L^p(\mu)$))
Definition 5 (Essential supremum norm ($L^\infty$ norm))
Definition 6 ($\mathcal{L}^\infty$ space ($\mathcal{L}^\infty(S, \mathcal{S}, \mu)$))
Definition 7 ($L^\infty$ space ($L^\infty(S, \mathcal{S}, \mu)$, $L^\infty(\mu)$))

\(L^p\) Spaces

0.1 \(L^p\) spaces

Definitions on this page