The L-p Spaces

The Lebesgue integral measures the size of a function, and there is one such measure per exponent $p$ , with a function small when its $p$ -th power has small integral. Each choice gives a complete normed space, a Banach space, and the single value $p=2$ gives the Hilbert space $L^2$ the geometry of the subject lives in. This post proves the two inequalities that make the $p$ -norm a norm and the completeness that makes the space Banach, on a fixed measure space $(\Omega,\mathcal A,\mu)$ [1]. For $1<p<\infty$ the conjugate exponent $q$ satisfies $\tfrac1p+\tfrac1q=1$ (equivalently $q=p/(p-1)$ ); the case $p=1$ needs no conjugate exponent and is handled separately.

#The p-norm

Definition1

For $1\le p<\infty$ , the space $L^p(\mu)$ consists of the measurable functions with $\int\abs f^p\,d\mu<\infty$ , with two identified when they agree almost everywhere, normed by

\norm f_p=\Big(\int\abs f^p\,d\mu\Big)^{1/p}. \tag{1}

Positivity and homogeneity of $\norm{\cdot}_p$ are immediate, and the null-set identification makes $\norm f_p=0$ force $f=0$ . The one nontrivial axiom is the triangle inequality, and it follows from a pointwise inequality between products and powers.

#Young, Holder, and Minkowski

Lemma2

For nonnegative reals $a,b$ and conjugate exponents $p,q>1$ , $ab\le\dfrac{a^p}{p}+\dfrac{b^q}{q}$ .

Proof

If $a$ or $b$ is zero the inequality is trivial, so take $a,b>0$ . The logarithm is concave, so the value at a weighted average is at least the weighted average of the values,

\log\Big(\frac{a^p}{p}+\frac{b^q}{q}\Big)\ge\frac1p\log a^p+\frac1q\log b^q=\log a+\log b=\log(ab), \tag{2}

using the weights $\tfrac1p,\tfrac1q$ summing to $1$ and the identities $\tfrac1p\log a^p=\log a$ . Exponentiating, which is monotone, gives $ab\le\frac{a^p }{p}+\frac{b^q}{q}$ .

Theorem3

For $p,q$ conjugate with $1<p<\infty$ and measurable $f,g$ , $\int\abs{fg}\,d\mu\le\norm f_p\,\norm g_q$ .

Proof

If $\norm f_p$ or $\norm g_q$ is $0$ , then $fg=0$ almost everywhere and the bound holds; if either is infinite it is trivial. So assume both are finite and positive. Apply Young's inequality pointwise to $a=\abs f/\norm f_p$ and $b=\abs g/\norm g_q$ ,

\frac{\abs{fg}}{\norm f_p\norm g_q}\le\frac{\abs f^p}{p\,\norm f_p^p}+\frac{\abs g^q}{q\,\norm g_q^q}, \tag{3}

and integrate. The right side integrates to $\tfrac1p+\tfrac1q=1$ , since $\int\abs f^p=\norm f_p^p$ and $\int\abs g^q=\norm g_q^q$ , so $\int\abs{fg}\,d\mu\le\norm f_p\norm g_q$ .

The case $p=q=2$ is the Cauchy-Schwarz inequality for the integral inner product. Holder's inequality in turn delivers the triangle inequality.

Theorem4

For $1\le p<\infty$ and $f,g\in L^p(\mu)$ , $\norm{f+g}_p\le\norm f_p+\norm g_p$ .

Proof

The case $p=1$ is the integral of $\abs{f+g}\le\abs f+\abs g$ . For $p>1$ , first $f+g\in L^p$ because $\abs{f+g}^p\le 2^{p-1}(\abs f^p+\abs g^p)$ by convexity of $t\mapsto t^p$ . If $\norm{f+g}_p=0$ there is nothing to prove, so assume it is positive. Split the integrand and apply Holder with the conjugate exponent $q=p/(p-1)$ to each piece,

\norm{f+g}_p^p=\int\abs{f+g}\,\abs{f+g}^{p-1}\,d\mu\le\big(\norm f_p+\norm g_p\big)\,\big\||f+g|^{p-1}\big\| _q, \tag{4}

using $\int\abs f\,\abs{f+g}^{p-1}\le\norm f_p\norm{|f+g|^{p-1}}_q$ and the same for $g$ . The last factor is $\big(\int\abs{f+g}^{(p-1)q}\big)^{1/q}=\big(\int\abs{f+g}^p\big)^{1/q}=\norm{f+g}_p^{p-1}$ , since $(p-1)q=p$ and $p/q=p-1$ . Dividing Equation (4) by the finite nonzero $\norm{f+g}_p^{p-1}$ leaves $\norm{f+g}_p\le\norm f_p+\norm g_p$ .

So $\norm{\cdot}_p$ is a genuine norm and $L^p(\mu)$ is a normed vector space. The remaining property, the one that makes it useful, is completeness.

#Completeness

Theorem5

For $1\le p<\infty$ , $L^p(\mu)$ is complete, hence a Banach space.

Proof

Let $(f_n)$ be Cauchy in $L^p$ . Choose a subsequence with $\norm{f_{n_{k+1}}-f_{n_k}}_p\le 2^{-k}$ , possible because the sequence is Cauchy, and set $g=\sum_{k}\abs{f_{n_{k+1}}-f_{n_k}}$ . Induction on Minkowski gives $\norm{\sum_{k=1}^N h_k}_p\le\sum_{k=1}^N\norm{h_k}_p$ , so with $h_k=\abs{f_{n_{k+1}}-f_{n_k}}$ every partial sum has $p$ -norm at most $\sum_k 2^{-k}\le 1$ , and the monotone convergence theorem applied to the increasing $g^p$ gives $\norm g_p\le 1$ , and in particular $g<\infty$ almost everywhere. Where $g$ is finite the telescoping series $f_{n_1}+\sum_k(f_{n_{k+1}}-f_{n_k})$ converges absolutely to a limit $f$ , and we set $f=0$ on the null set $\{g=\infty\}$ , so $f$ is measurable everywhere as an a.e. limit of measurable functions, with $\abs f\le\abs{f_{n_1}}+g\in L^p$ a.e., so $f\in L^p$ . Each partial sum gives $\abs{f_{n_k}}\le\abs{f_{n_1}}+g$ , so $\abs{f-f_{n_k}} \le 2\abs{f_{n_1}}+2g$ a.e., an integrable dominator independent of $k$ since $f_{n_1},g\in L^p$ , and as $f_{n_k}\to f$ almost everywhere the dominated convergence theorem gives $\norm{f-f_{n_k}}_p\to 0$ . A Cauchy sequence with a convergent subsequence converges to the same limit, so $f_n\to f$ in $L^p$ .

Completeness is what guarantees an $L^p$ limit of functions is again a function rather than a formal object. For $p=2$ it is the completeness that makes $L^2$ a Hilbert space, the only exponent at which the norm comes from an inner product, by the parallelogram law. The other exponents lose the geometry but keep the completeness. Holder's inequality drives the moment inequalities of probability and the interpolation theorems of harmonic analysis. It also yields the duality that identifies the dual of $L^p$ with $L^q$ , the structural fact that makes the spaces reflexive for $1<p<\infty$ . The Banach space of integrable $p$ -th powers is the natural habitat of quantitative analysis.

[1]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed. Wiley, 1999.

Explore connections

see in the atlas

referenced by (1)

Uniform Integrability and the Vitali Theorem

cite

@misc{lp-spaces,
  author = {Zac Kienzle},
  title  = {The L-p Spaces},
  year   = {2026},
  month  = {06},
  url    = {https://zackienzle.com/blog/lp-spaces}
}

#The p-norm

Definition1

For $1\le p<\infty$ , the space $L^p(\mu)$ consists of the measurable functions with $\int\abs f^p\,d\mu<\infty$ , with two identified when they agree almost everywhere, normed by

\norm f_p=\Big(\int\abs f^p\,d\mu\Big)^{1/p}. \tag{1}

#Young, Holder, and Minkowski

Lemma2

For nonnegative reals $a,b$ and conjugate exponents $p,q>1$ , $ab\le\dfrac{a^p}{p}+\dfrac{b^q}{q}$ .

Proof

If $a$ or $b$ is zero the inequality is trivial, so take $a,b>0$ . The logarithm is concave, so the value at a weighted average is at least the weighted average of the values,

\log\Big(\frac{a^p}{p}+\frac{b^q}{q}\Big)\ge\frac1p\log a^p+\frac1q\log b^q=\log a+\log b=\log(ab), \tag{2}

using the weights $\tfrac1p,\tfrac1q$ summing to $1$ and the identities $\tfrac1p\log a^p=\log a$ . Exponentiating, which is monotone, gives $ab\le\frac{a^p }{p}+\frac{b^q}{q}$ .

Theorem3

For $p,q$ conjugate with $1<p<\infty$ and measurable $f,g$ , $\int\abs{fg}\,d\mu\le\norm f_p\,\norm g_q$ .

Proof

\frac{\abs{fg}}{\norm f_p\norm g_q}\le\frac{\abs f^p}{p\,\norm f_p^p}+\frac{\abs g^q}{q\,\norm g_q^q}, \tag{3}

and integrate. The right side integrates to $\tfrac1p+\tfrac1q=1$ , since $\int\abs f^p=\norm f_p^p$ and $\int\abs g^q=\norm g_q^q$ , so $\int\abs{fg}\,d\mu\le\norm f_p\norm g_q$ .

The case $p=q=2$ is the Cauchy-Schwarz inequality for the integral inner product. Holder's inequality in turn delivers the triangle inequality.

Theorem4

For $1\le p<\infty$ and $f,g\in L^p(\mu)$ , $\norm{f+g}_p\le\norm f_p+\norm g_p$ .

Proof

\norm{f+g}_p^p=\int\abs{f+g}\,\abs{f+g}^{p-1}\,d\mu\le\big(\norm f_p+\norm g_p\big)\,\big\||f+g|^{p-1}\big\| _q, \tag{4}

So $\norm{\cdot}_p$ is a genuine norm and $L^p(\mu)$ is a normed vector space. The remaining property, the one that makes it useful, is completeness.

#Completeness

Theorem5

For $1\le p<\infty$ , $L^p(\mu)$ is complete, hence a Banach space.

Proof

[1]

G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed. Wiley, 1999.

Explore connections

see in the atlas

referenced by (1)

Uniform Integrability and the Vitali Theorem

cite

@misc{lp-spaces,
  author = {Zac Kienzle},
  title  = {The L-p Spaces},
  year   = {2026},
  month  = {06},
  url    = {https://zackienzle.com/blog/lp-spaces}
}