\(L^p\) Spaces
Last modified: 2026-10-31 15:22:31 (PDT)
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
\[ \begin{aligned} \mathopen{}\left\lVert f\right\rVert\mathclose{}_2 &= \mathopen{}\left(\int_0^1 x^2 \, dx\right)\mathclose{}^{1/2} \\ &= \mathopen{}\left(\frac{1}{3}\right)\mathclose{}^{1/2}. \end{aligned} \]
[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")[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()
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.
[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")[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()
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, \(\mathopen{}\left\lVert f\right\rVert\mathclose{}_\infty = 1\).
[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)")[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()
