L^p Spaces

Norms on functions, and why membership depends on the exponent.

The idea

The $L^{p}$ spaces equip functions with a norm, so that the distance $\lVert f - g \rVert_{p}$ between two functions — and with it convergence and approximation — makes sense. The norm measures a function by an integral over its whole domain, not by a largest value that a single point could decide.

Fix a real number $p$ with $1 \le p < \infty$. The $L^{p}$ norm of a measurable function $f$ is $\lVert f \rVert_{p} = \left( \int |f|^{p} \right)^{1/p},$ and $L^{p}$ is the collection of $f$ for which this integral is finite. The outer root makes the norm scale linearly: $\lVert cf \rVert_{p} = |c| \, \lVert f \rVert_{p}$. Raising $|f|$ to the power $p$ magnifies the values above $1$ and shrinks the values below $1$, so one function can belong to $L^{p}$ for some exponents and not others. The two threats to membership are a spike, where the function blows up at a point, and a tail, where the domain runs to infinity and the function must decay. For $p = \infty$ the norm is the essential supremum of $|f|$: the smallest $M$ with $|f(x)| \le M$ outside a set of measure zero.

One repair remains. A function that is nonzero only on a set of measure zero has norm $0$ without being the zero function, so we take the elements of $L^{p}$ to be equivalence classes of functions that agree almost everywhere. With that identification $\lVert \cdot \rVert_{p}$ is a genuine norm.

Ways to work on it

• Walkthrough. Define the L^p norms and the spaces of functions they measure.
• Proof. Holder's inequality from Young's, with Cauchy-Schwarz as the symmetric case.
• Practice. Decide whether a power function belongs to a given L^p.
• Hardest. Compare membership in L^1 and L^2, and put Holder's inequality to work.

Not sure where to start? Take the ten-question placement test.