Real-Analysis Normed Linear-Spaces, guaranteed and verified 100% Pass

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Normed Linear Spaces


Let 𝐸 be a measurable set of real numbers.

Let 𝐹 be the collection of measurable extended real valued function on 𝐸 that
are finite a.e. on 𝐸.

Let’s define two functions in 𝐹 to be equivalent, 𝑓 ≅ 𝑔, if 𝑓 (𝑥 ) = 𝑔(𝑥) for
almost all 𝑥 ∈ 𝐸.

≅ is an equivalence relation (i.e. it’s reflexive, symmetric, and transitive).
So we can partition 𝐹 into 𝐹⁄≅ , equivalence classes of functions.


Notice that 𝐹⁄≅ is a linear space: 𝛼𝑓 + 𝛽𝑔 ∈ 𝐹 if 𝑓, 𝑔 ∈ 𝐹 and

𝛼, 𝛽 ∈ ℝ, and the zero element is the class of functions that are 0 a.e. on 𝐸.


Def. 𝑳𝒑 (𝑬) = {𝑓 ∈ 𝐹⁄≅ | ∫𝐸 |𝑓|𝑝 < ∞}; 1 ≤ 𝑝 < ∞.


Notice that 𝐿𝑝 (𝐸 ) is a linear subspace of 𝐹⁄≅ since for any 𝑎, 𝑏 ∈ ℝ:

|𝑎 + 𝑏| ≤ |𝑎| + |𝑏| ≤ 2max {|𝑎|, |𝑏|}
So |𝑎 + 𝑏|𝑝 ≤ 2𝑝 (max{|𝑎|, |𝑏|})𝑝 ≤ 2𝑝 (|𝑎|𝑝 + |𝑏|𝑝 ).

Thus if 𝑓, 𝑔 ∈ 𝐿𝑝 (𝐸 ) then 𝛼𝑓 ∈ 𝐿𝑝 (𝐸 ) because

∫𝐸 |𝛼𝑓|𝑝 = |𝛼|𝑝 ∫𝐸 |𝑓|𝑝 < ∞.

And 𝑓 + 𝑔 ∈ 𝐿𝑝 (𝐸 ) because |𝑓 + 𝑔|𝑝 ≤ 2𝑝 (|𝑓 |𝑝 + |𝑔|𝑝 )

So ∫𝐸 |𝑓 + 𝑔|𝑝 ≤ ∫𝐸 2𝑝 (|𝑓 |𝑝 + |𝑔|𝑝 ) = 2𝑝 (∫𝐸 |𝑓 |𝑝 + ∫𝐸 |𝑔|𝑝 ) < ∞.

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So 𝛼𝑓 + 𝛽𝑔 ∈ 𝐿𝑝 (𝐸 ).


Clearly 𝑓 = 0 a.e. on 𝐸 is also in 𝐿𝑝 (𝐸 ).

Notice that 𝐿1 (𝐸 ) is just the integrable functions over 𝐸.



Def. 𝑓 ∈ 𝐹 is called essentially bounded if there is come 𝑀 ≥ 0 such that
|𝑓(𝑥 )| ≤ 𝑀 for almost all 𝑥 ∈ 𝐸. 𝑀 is called an essential upper bound for 𝑓.


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Ex. 𝑓 (𝑥 ) = if 𝑥 ∈ ℚ, 𝑥 ≠ 0
𝑥
= 2 if 𝑥 ∉ ℚ, 𝑥 ≠ 0
Is essentially bounded on (−∞, ∞) because |𝑓 (𝑥 )| ≤ 2 a.e..


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Ex. 𝑓(𝑥 ) = if 𝑥 ∉ ℚ, 𝑥 ≠ 0
𝑥
= 2 if 𝑥 ∈ ℚ, 𝑥 ≠ 0
Is not essentially bounded on (−∞, ∞).



Def. 𝑳∞ (𝑬) is the set of essentially bounded function on 𝐸.



𝐿∞ (𝐸 ) is also a linear subspace of 𝐹⁄≅.


Def. Real valued functions whose domain is a linear space such as 𝐿𝑝 (𝐸 ) are
called functionals.