What is the definition of the p-norm for a measurable function when $0 < p < +\infty$?

Conditions

• E is a measurable subset of $\mathbb{R}^n$.
• f is a measurable function on E.
• $0 < p < +\infty$.

Reasoning, step by step

1. Identify the measurable set E and the measurable function f.
2. Take the absolute value of the function, $|f(x)|$.
3. Raise the absolute value to the power of p, yielding $|f(x)|^p$.
4. Integrate this expression over the set E.
5. Raise the resulting integral to the power of $1/p$.

Example

The video explicitly writes the formula on the board: $\|f\|_p := \left( \int_E |f(x)|^p dx \right)^{\frac{1}{p}}$.

Common misconceptions

• Assuming the p-norm formula is immediately a verified norm; the video clarifies it is a formal definition and its norm axioms must be verified later.
• Forgetting to take the absolute value before raising to the power of p, which is necessary for the integral to be well-defined for real-valued functions.

Watch the explanation

Explore next

• How is the $L^p$ space defined using the p-norm?
• What is the definition of the infinity norm when $p = +\infty$?