Real-Analysis Bounded Linear Functionals on Lp Spaces, guaranteed and verified 100% Pass

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Bounded Linear Functionals on 𝐿𝑝 Spaces


Def. A linear functional on a linear space 𝑋 is a real valued function 𝑇 on 𝑋 such
that for 𝑔, ℎ ∈ 𝑋 and 𝛼, 𝛽 ∈ ℝ

𝑇(𝛼𝑔 + 𝛽ℎ) = 𝛼𝑇(𝑔) + 𝛽𝑇(ℎ).

Notice that if 𝑇, 𝑆 are linear functionals on 𝑋, so is 𝑎𝑇 + 𝑏𝑆, 𝑎, 𝑏 ∈ ℝ.

Since the functional 𝑇: 𝑋 → ℝ defined by 𝑇(𝑔) = 0 for all 𝑔 ∈ 𝑋 is linear, the
set of linear functionals on 𝑋 is itself a linear space.



Ex. Let 𝐸 be a measurable set, 1 ≤ 𝑝 < ∞ and 𝑞 the conjugate of 𝑝. If
𝑔 ∈ 𝐿𝑞 (𝐸) then define:
𝑇: 𝐿𝑝 (𝐸) → ℝ

by 𝑇(𝑓) = ∫𝐸 𝑓𝑔; 𝑓 ∈ 𝐿𝑝 (𝐸).
By the Holder inequality 𝑓𝑔 ∈ 𝐿1 (𝐸).

𝑇 is linear because integration over 𝐸 is linear. Notice also that:

|𝑇(𝑓)| = | ∫𝐸 𝑓𝑔 | ≤ ∫𝐸 |𝑓𝑔| ≤ ‖𝑓‖𝑝 ‖𝑔‖𝑞
For all 𝑓 ∈ 𝐿𝑝 (𝐸).



Def. For a normed linear space 𝑋, a linear functional is said to be bounded if
there is an 𝑀 ≥ 0 such that:

|𝑇(𝑓 )| ≤ 𝑀‖𝑓‖ for all 𝑓 ∈ 𝑋.
The infimum of all such 𝑀 is called the norm of 𝑇, denoted ‖𝑇‖∗ .

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Ex. 𝑇: 𝐿𝑝 (𝐸) → ℝ by 𝑇(𝑓 ) = ∫𝐸 𝑓𝑔 ; 𝑓 ∈ 𝐿𝑝 (𝐸), with a fixed
𝑔 ∈ 𝐿𝑞 (𝐸), is a bounded linear functional since

|𝑇(𝑓)| ≤ ∫𝐸 |𝑓𝑔| ≤ ‖𝑓‖𝑝 ‖𝑔‖𝑞 , (𝑀 = ‖𝑔‖𝑞 ).


Let 𝑇 be a bounded linear functional on 𝑋, and 𝑀 = ‖𝑇‖∗ . Then for any
𝑓, ℎ ∈ 𝑋:
|𝑇(𝑓) − 𝑇(ℎ)| = |𝑇(𝑓 − ℎ)| ≤ ‖𝑇‖∗ ‖𝑓 − ℎ‖.

Thus if 𝑓𝑛 → 𝑓 in 𝑋 then :

|𝑇(𝑓𝑛 ) − 𝑇(𝑓)| ≤ ‖𝑇‖∗ ‖𝑓𝑛 − 𝑓‖.

So if lim ‖𝑓𝑛 − 𝑓 ‖ = 0 then lim |𝑇 (𝑓𝑛 ) − 𝑇(𝑓 )| = 0.
𝑛→∞ 𝑛→∞

That is, if 𝑇 is a bounded linear functional on 𝑋 and 𝑓𝑛 → 𝑓 in 𝑋 then the
sequence of real numbers {𝑇(𝑓𝑛 )} converges to 𝑇(𝑓) in ℝ, ie 𝑇 is a continuous
map from 𝑋 to ℝ.



Because 𝑇 is linear
‖𝑇‖∗ = inf{𝑀| |𝑇(𝑓)| ≤ 𝑀‖𝑓‖, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑓 ∈ 𝑋}
= sup{𝑇(𝑓 )| 𝑓 ∈ 𝑋, ‖𝑓‖ ≤ 1}
𝑓
Since if 𝑓 ≠ 0 then |𝑇 (
‖𝑓‖
)| ≤ 𝑀 ⟺ |𝑇(𝑓)| ≤ 𝑀‖𝑓‖.



Prop. Let 𝑋 be a normed linear space. Then the collection of bounded linear
functionals on 𝑋 is a linear space on which ‖ ‖∗ is a norm. This normed linear
space is called the dual space of 𝑿 and is denoted 𝑋 ∗ .