TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-23 (gpt-6-sol)

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$L^{p}$ and $L^{\infty}$ are vector spaces for $p \geq 1$

Statement

Let $(X, A, μ)$ be a measure space.

For each $1 \leq p < \infty$ , the class $L^{p} (μ)$ is a real vector space under pointwise addition and scalar multiplication.
The class $L^{\infty} (μ)$ is a real vector space under the same operations.

Facts & Assumptions

Given: A measure space $(X, A, μ)$ .

[L1]

$L^{p} (μ)$ and $L^{\infty} (μ)$ are the classes defined in The function space $L^{p} (μ)$ for $0 < p < \infty$ and The space $L^{\infty} (μ)$ of essentially bounded measurable functions.

[L2]

Sums, scalar multiples, and absolute values of measurable real-valued functions are measurable (Closure properties of measurable functions used by the integral).

[L3]

The nonnegative integral is monotone, additive, and homogeneous for positive finite scalars; the zero function has integral zero (Additivity of the nonnegative Lebesgue integral, Monotonicity and nonnegative homogeneity of the nonnegative integral).

[L4]

A finite essential supremum is an attained essential bound (The essential supremum is attained as the least essential bound).

[L5]

Countable unions of measurable null sets are measurable and null (Finite and countable subadditivity of measures).

[L6]

A vector space over $R$ means the structure defined in Vector space over a field.

Proof

technique · direct

1.1L1L2L3givenalgebra

Fix $1 \leq p < \infty$ and let $f, g \in L^{p} (μ)$ and $a \in R$ . Measurability of $f + g$ and $a f$ follows from [L2]. Pointwise, $∣ f + g ∣^{p} \leq (2 \max {∣ f ∣, ∣ g ∣})^{p} \leq 2^{p} (∣ f ∣^{p} + ∣ g ∣^{p}) .$ By [L3], $\int ∣ f + g ∣^{p} d μ \leq 2^{p} (\int ∣ f ∣^{p} d μ + \int ∣ g ∣^{p} d μ) < \infty .$ If $a \neq 0$ , positive homogeneity in [L3] gives $\int ∣ a f ∣^{p} d μ = ∣ a ∣^{p} \int ∣ f ∣^{p} d μ < \infty$ ; if $a = 0$ , the integral is zero without forming $0 \cdot \infty$ . Thus $f + g, a f \in L^{p} (μ)$ . The zero function belongs to the class, and additive inverses use $a = - 1$ .

1.2

Let $f, g \in L^{\infty} (μ)$ with $M : = ∥ f ∥_{\infty}$ and $N : = ∥ g ∥_{\infty}$ . There are measurable null sets $E_{f}, E_{g}$ such that $∣ f ∣ \leq M$ on $X ∖ E_{f}$ and $∣ g ∣ \leq N$ on $X ∖ E_{g}$ . With $E : = E_{f} \cup E_{g}$ , $E$ is measurable and null, and on $X ∖ E$ one has [L1, L2, L4, L5, given] $∣ f + g ∣ \leq ∣ f ∣ + ∣ g ∣ \leq M + N, ∣ a f ∣ = ∣ a ∣ ∣ f ∣ \leq ∣ a ∣ M .$ Thus $f + g$ and $a f$ are essentially bounded; measurability again comes from [L2].

2.1step 1.1L6

The pointwise addition and scalar-multiplication identities are inherited from real-valued functions. Hence [L6] makes $L^{p} (μ)$ a real vector space for every $1 \leq p < \infty$ .

3.1step 1.2L6∎

The pointwise identities are again inherited from real-valued functions, so [L6] makes $L^{\infty} (μ)$ a real vector space.

$L^{p}$ and $L^{\infty}$ are vector spaces for $p \geq 1$

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

LpLp and L∞L∞ are vector spaces for p≥1p≥1

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

$L^{p}$ and $L^{\infty}$ are vector spaces for $p \geq 1$