Lp Spaces - Examples and Constructions

Understanding concrete examples of $L^p$ spaces and their relationships helps develop intuition for functional analysis. Different choices of $p$ capture different aspects of function behavior.

ExampleComparing Lp Norms

Consider $f(x) = x^{-\alpha}$ on $(0, 1)$ with Lebesgue measure. When is $f \in L^p$ ?

$\|f\|_p^p = \int_0^1 x^{-\alpha p} \, dx$

This integral converges at $0$ if and only if $-\alpha p > -1$ , i.e., $\alpha < 1/p$ . Thus: $f \in L^p((0,1)) \Leftrightarrow \alpha < \frac{1}{p}$

For $\alpha = 1/2$ :

$f \in L^1$ : Yes, since $1/2 < 1$
$f \in L^2$ : No, since $1/2 = 1/2$
$f \in L^p$ for $p < 2$ : Yes

This illustrates how larger $p$ requires better decay for integrability.

ExampleSequence Spaces

For $\ell^p$ (sequences with $\sum |a_n|^p < \infty$ ):

$\ell^2$ : The space of square-summable sequences. Example: $a_n = 1/n$ is NOT in $\ell^2$ since $\sum 1/n^2 = \pi^2/6 < \infty$ but $a_n = 1/n^{1/2}$ is NOT.
$\ell^1$ : Absolutely summable sequences. Example: $a_n = 1/n^2 \in \ell^1$ , but $a_n = 1/n \notin \ell^1$ .
$\ell^\infty$ : Bounded sequences. Example: $a_n = (-1)^n \in \ell^\infty$ with $\|a\|_\infty = 1$ .
Inclusion: $\ell^1 \subset \ell^2 \subset \ell^\infty$ (strict inclusions).

Remark

Construction techniques for $L^p$ functions:

Truncation: Given $f \in L^p$ , define $f_n = f \chi_{\{|f| \leq n\}}$ . Then $f_n \to f$ in $L^p$ norm. This allows approximation by bounded functions.
Mollification: On $\mathbb{R}^n$ , convolving $f \in L^p$ with a smooth approximation to the identity $\phi_\epsilon$ gives $f * \phi_\epsilon \in C^\infty \cap L^p$ with $\|f * \phi_\epsilon - f\|_p \to 0$ .
Simple function approximation: Every $f \in L^p$ is the $L^p$ limit of simple functions.
Continuous function approximation: For $1 \leq p < \infty$ , continuous functions with compact support are dense in $L^p(\mathbb{R}^n)$ .

The flexibility of $L^p$ spaces makes them natural settings for diverse applications: $L^1$ for probability densities, $L^2$ for quantum mechanics and signal processing, $L^\infty$ for control theory and optimization. The choice of $p$ depends on the problem structure and desired properties.