Lp Spaces and the Inequalities They Rest On
Functions with integrable p-th power form a normed space once functions that agree almost everywhere are treated as equal. Holder's and Minkowski's inequalities are what make the norm behave like a length.
What a learner can do afterwards
- Use Holder's inequality to bound the integral of a product
- Explain the identification of functions agreeing almost everywhere and why the norm forces it
- Say what makes the case p equal to 2 different from every other p
1 · Read
You measure a function by its Lp norm: integrate its p-th power and take the p-th root. Minkowski proved the triangle inequality for this measurement, which is why the space of functions with finite norm behaves like a geometry with lengths. Holder bounds the integral of a product by the product of the norms, using conjugate exponents whose reciprocals sum to 1.
The indicator of the interval from 1 to 4 has L1 norm equal to its length, which is 3. The constant 3 on an interval of length 4 has L2 norm equal to the square root of 9 times 4, which is 6. You compute both directly from the definition of the norm.
You must treat two functions that agree except on a null set as the same element of Lp. The reason is definiteness: the indicator of a single point is nonzero somewhere yet its integral is 0, so without this step a nonzero function would carry norm 0. Identifying almost everywhere equal functions repairs the norm.
When p equals 2, the norm comes from an inner product, so you gain orthogonality and the parallelogram law, and the space is a Hilbert space. For every other exponent no inner product sits behind the norm. As a quick check for Holder, the conjugate of 4 is 4 over 3, since 1 over 4 plus 3 over 4 equals 1.
Holder and Minkowski make Lp a normed geometry, almost everywhere equal functions count as one element, and only p equal to 2 brings an inner product.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.