Lᵖ Spaces: Notation

(i) Base. This setting extends Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background and Real Normed, Inner Product and Hilbert Spaces: Notation: their conventions and notation are in force; here $p,q$ are real numbers with $p,q\ge1$, overriding the natural numbers $p,q$ of Measurable Functions and the Lebesgue Integral: Notation (i).

(ii) The spaces $L^{p}(\mu)$. Real powers $t^{a}$ are as defined there. $|f|^{p}$, $\mathcal{L}^{p}(\mu)$ and $\|f\|_{p}$ are as in Power-Integrable Functions and the p-Seminorm (i), Power-Integrable Functions and the p-Seminorm (ii) and Power-Integrable Functions and the p-Seminorm (iii). $L^{p}(\mu)$, the classes $[f]$, the sum and scalar multiples on $L^{p}(\mu)$, its norm and, for $p=2$, its inner product are as in The Lebesgue Space Lᵖ(μ) (i), The Lebesgue Space Lᵖ(μ) (ii), The Lebesgue Space Lᵖ(μ) (iii) and The Lebesgue Space Lᵖ(μ) (iv). An element of $L^{p}(\mu)$ is written by a representative, $f$ for $[f]$, where a statement does not depend on the representative. Background: Properties of Real Powers, Real Powers of the Absolute Value of a Measurable Function Are Measurable, Conjugate Exponents and Young's Inequality, Elementary Properties of the p-Seminorm, Hölder's Inequality, Minkowski's Inequality, Almost-Everywhere Equality on the p-Integrable Functions, The Riesz–Fischer Theorem and L²(μ) Is a Real Hilbert Space.

(iii) The spaces $L^{2}(\mu;\mathbb{R}^{n})$. $|f|^{2}$, for measurable $f:X\to\mathbb{R}^{n}$, and $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ are as in Square-Integrable Vector-Valued Functions (i) and Square-Integrable Vector-Valued Functions (ii). $L^{2}(\mu;\mathbb{R}^{n})$, the classes $[f]$, its sum and scalar multiples and its inner product are as in The Lebesgue Space L²(μ;ℝⁿ) (i), The Lebesgue Space L²(μ;ℝⁿ) (ii) and The Lebesgue Space L²(μ;ℝⁿ) (iii), and its elements are written by representatives as in (ii). Background: Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality and L²(μ;ℝⁿ) Is a Real Hilbert Space.

(iv) Closed intervals. For $a,b\in\mathbb{R}$ with $a<b$, $\mathcal{L}^{p}([a,b])$, $L^{p}([a,b])$, $\mathcal{L}^{2}([a,b];\mathbb{R}^{n})$ and $L^{2}([a,b];\mathbb{R}^{n})$ denote $\mathcal{L}^{p}(\lambda_{[a,b]})$, $L^{p}(\lambda_{[a,b]})$, $\mathcal{L}^{2}(\lambda_{[a,b]};\mathbb{R}^{n})$ and $L^{2}(\lambda_{[a,b]};\mathbb{R}^{n})$, for $\lambda_{[a,b]}$ as in Measurable Functions and the Lebesgue Integral: Standing Notation (vii).