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The parallelogram law on two indicator functions in $L^{2}$

Example

On $[0, 1]$ with Lebesgue measure, let

$f : = χ_{[0, 1 / 2]}, g : = χ_{(1 / 2, 1]} .$

Then

$∥ f + g ∥_{2}^{2} + ∥ f - g ∥_{2}^{2} = 2 ∥ f ∥_{2}^{2} + 2 ∥ g ∥_{2}^{2} = 2.$

Facts & Assumptions

Given: The two indicator functions $f$ and $g$ above.

[L1]

The $L^{2}$ parallelogram law has already been proved (The parallelogram law in $L^{2}$ ).

Verification

Proof technique: Evaluate all four $L^{2}$ norms on a pair of simple indicator functions and compare the two sides directly.

1.1

Since $f$ and $g$ are disjoint indicators of sets of measure $1 / 2$ , [given, algebra] $∥ f ∥_{2}^{2} = ∥ g ∥_{2}^{2} = \frac{1}{2}, ∥ f + g ∥_{2}^{2} = 1, ∥ f - g ∥_{2}^{2} = 1.$

2.1

Therefore [L1, step 1.1] $∥ f + g ∥_{2}^{2} + ∥ f - g ∥_{2}^{2} = 1 + 1 = 2 = 2 \cdot \frac{1}{2} + 2 \cdot \frac{1}{2} = 2 ∥ f ∥_{2}^{2} + 2 ∥ g ∥_{2}^{2},$ exactly as [L1] predicts. ∎

Depends on

The parallelogram law in $L^2$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Measure Theory, Chapter 15 (standard reference, not scraped)

The parallelogram law on two indicator functions in L2L2