ell^p

Let $𝔽$ be either $ℝ$ or $ℂ$, and let $p\in ℝ$ with $p\ge 1$. We define ${\mathrm{\ell }}^p$ to be the set of all sequences ${(a_i)}_{i\ge 0}$ in $𝔽$ such that

$$\sum _{i=0}^{\mathrm{\infty }}{|a_i|}^p$$

converges.

We also define ${\mathrm{\ell }}^{\mathrm{\infty }}$ to be the set of all bounded (http://planetmath.org/BoundedInterval) sequences ${(a_i)}_{i\ge 0}$ with norm given by

$${\parallel (a_i)\parallel }_{\mathrm{\infty }}=\sup \{|a_i|:i\ge 0\}.$$

By defining addition and scalar multiplication pointwise, ${\mathrm{\ell }}^p(𝔽)$ and${\mathrm{\ell }}^{\mathrm{\infty }}(𝔽)$ have a natural vector space stucture.That the sum of two elements on ${\mathrm{\ell }}^p(𝔽)$ is again an elementin ${\mathrm{\ell }}^p(𝔽)$ follows from Minkowski inequality(see below).We can make ${\mathrm{\ell }}^p$ into a normed vector space, by defining the norm as

$${\parallel (a_i)\parallel }_p={(\sum _{i=0}^{\mathrm{\infty }}{|a_i|}^p)}^{1/p}.$$

The normed vector spaces ${\mathrm{\ell }}^{\mathrm{\infty }}$ and ${\mathrm{\ell }}^p$ for $p\ge 1$ are complete under these norms, making them into Banach spaces. Moreover, ${\mathrm{\ell }}^2$ is a Hilbert space under the inner product

$$⟨(a_i),(b_i)⟩=\sum _{i=0}^{\mathrm{\infty }}{a}_i\overline{b_i}$$

where $\overline{x}$ denotes the complex conjugate of $x$.

For $p>1$ the (continuous) dual space of ${\mathrm{\ell }}^p$ is ${\mathrm{\ell }}^q$ where $\frac{1}{p}+\frac{1}{q}=1$, and the dual space of ${\mathrm{\ell }}^1$ is ${\mathrm{\ell }}^{\mathrm{\infty }}$.

Properties

1. If $a=(a_0,a_1,\mathrm{\dots })\in {\mathrm{\ell }}^p(𝔽)$ for , then${lim}_{k\to \mathrm{\infty }}{a}_k=0$.(proof. (http://planetmath.org/ThenA_kto0IfSum_k1inftyA_kConverges))

2. For , ${\mathrm{\ell }}^p(𝔽)$ is separable, and ${\mathrm{\ell }}^{\mathrm{\infty }}(𝔽)$is not separable.

3. Minkowski inequality. If $a,b\in {\mathrm{\ell }}^p(𝔽)$ where $p\ge 1$, then

   $${\parallel a+b\parallel }_p\le {\parallel a\parallel }_p+{\parallel b\parallel }_p.$$