Lp Spaces
Spaces of functions with integrable p-th power.
Key idea
Spaces of functions with integrable p-th power.
In depth
Lᵖ is the space of measurable functions with ∫|f|ᵖ < ∞, normed by ‖f‖ₚ = (∫|f|ᵖ)^(1/p) (functions equal almost everywhere are identified). They are complete, so Banach spaces; L² is a Hilbert space.
Worked example
f(x) = 1/√x is in L¹(0, 1) but not in L²(0, 1).
Formula
Real-world uses
- Measuring signal energy in L² and error in L¹ across audio, image and machine learning tasks
Common mistake
Forgetting that elements are equivalence classes, not individual functions.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Lp Spaces?
Spaces of functions with integrable p-th power. It is a topic at the expert level in Analysis & Topology.
What is the formula for Lp Spaces?
The key formula is ‖f‖ₚ = (∫|f|ᵖ)^(1/p).
Where is Lp Spaces used in real life?
Real-world examples: Measuring signal energy in L² and error in L¹ across audio, image and machine learning tasks.
What should I learn before Lp Spaces?
Make sure you are comfortable with Fubini’s Theorem, the previous topic in Analysis & Topology.
What is a common mistake with Lp Spaces?
Forgetting that elements are equivalence classes, not individual functions.
How can I practise Lp Spaces?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Lp Spaces?
Move on to Hölder and Minkowski Inequalities next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.