𝓁^𝑝 Spaces - Lebesgue Spaces (𝓁^𝑝)

𝓁^𝑝 spaces - Types

For 0<𝑝<∞, 𝓁^𝑝 is a linear subspace of vector space of all sequences 𝐹ℕ consisting of all sequences 𝑥̅= (𝑥_𝑛)_𝑛∈ℕ satisfying:

• ${\sum }_n|x_n{|}^p<\infty$

If 𝑝≥1, then the real-valued function ||·||𝑝 on 𝓁^𝑝 defined by:

• $||x|{|}_p=({\sum }_n|x_n{|}^p{)}^{1/p}forallx∊{𝓁}^p$

defines a norm on 𝓁^𝑝. In fact, 𝑙^𝑝is a complete metric space with respect to this norm and therefore is a Banach space.

If 𝑝=2 then 𝓁² is also a Hilbert space when endowed with its canonical inner product, called the Euclidean inner product, defined for all 𝑥̅,𝑦̅∊𝓁^𝑝 by:

• $⟨\overline{x},\overline{y}⟩={\sum }_n\overline{x_n}{y}_n$

The canonical norm induced by this inner product is the usual 𝐿2-norm:

• $||x|{|}_2=\sqrt{⟨x,x⟩}forallx∊{𝓁}^p$

If 𝑝=∞, then 𝓁^∞ is defined to be the space of all bounded sequences endowed with the norm:

• $||x|{|}_{\infty }=su{p}_n|x_n|$

𝓁^∞ is also a Banach space

If 0<𝑝<1 then 𝓁^𝑝 does not carry a norm, but rather a distance metric defined by:

• $d(x,y)={\sum }_n|x_n-y_n{|}^p$