Sequence Spaces & Lebesgue Spaces

Sequence and Lebesgue spaces are the main test bench for functional analysis. They look concrete enough for coordinate calculations but already exhibit the infinite-dimensional phenomena—noncompact balls, inequivalent norms, weak convergence, endpoint duality failures—that operator theory must handle.

The sequence spaces #

For $1\le p<\infty$,

$$\ell^p=\left\{x=(x_n):\sum_{n=1}^{\infty}|x_n|^p<\infty\right\}, \qquad \|x\|_p=\left(\sum_{n=1}^{\infty}|x_n|^p\right)^{1/p}.$$

Also

$$\ell^\infty=\{x:\sup_n|x_n|<\infty\},\qquad c_0=\{x:x_n\to0\},$$

both with the sup norm. Each is Banach. The finite-support sequences $c_{00}$ are dense in $\ell^p$ for $p<\infty$ and in $c_0$, but their closure in $\ell^\infty$ is only $c_0$.

For sequences, $1\le p<q\le\infty$ implies $\ell^p\subseteq\ell^q$ and $\|x\|_q\le\|x\|_p$. This direction is special to counting measure; do not transfer it blindly to function spaces.

Lebesgue spaces identify almost-everywhere equality #

On a measure space $(\Omega,\mathcal F,\mu)$,

$$L^p(\Omega)=\left\{f:\int_\Omega |f|^p\,d\mu<\infty\right\}/\text{a.e. equality}$$

for $p<\infty$, and

$$\|f\|_\infty=\mathop{\mathrm{ess\,sup}}\limits_{x\in\Omega}|f(x)|.$$

An $L^p$ element is an equivalence class, not a pointwise function. Evaluation $f\mapsto f(x_0)$ is therefore not generally well-defined. Every $L^p$ for $1\le p\le\infty$ is Banach (Riesz–Fischer).

Hölder creates the dual pairing #

If $1/p+1/q=1$, then

$$\int_\Omega |fg|\,d\mu\le\|f\|_p\|g\|_q.$$

Young's scalar inequality $ab\le a^p/p+b^q/q$, applied after normalizing $f$ and $g$, explains the conjugate exponents. Minkowski's inequality

$$\|f+g\|_p\le\|f\|_p+\|g\|_p$$

then proves that $\|\cdot\|_p$ is a norm. At $p=2$, Hölder is Cauchy–Schwarz and the norm comes from an inner product.

Inclusion depends on the measure space #

If $\mu(\Omega)<\infty$ and $1\le q<p\le\infty$, then

$$L^p(\Omega)\subseteq L^q(\Omega),\qquad \|f\|_q\le \mu(\Omega)^{1/q-1/p}\|f\|_p.$$

On an infinite-measure space, neither inclusion need hold. On $(0,\infty)$, power behavior near zero and infinity can be chosen so that a function lies in one $L^p$ but not another. The correct question is never simply “is $L^p\subset L^q$?”; it includes the underlying measure.

Structural map #

The exact $L^p$ separability and endpoint-duality statements depend on the measure space. For a σ-finite measure space, the familiar $(L^1)^*=L^\infty$ representation holds; the dual of $L^\infty$ contains finitely additive functionals and is not merely $L^1$.

Norm, pointwise, and weak convergence differ #

On $[0,1]$, let $f_n=n^{1/p}\mathbf 1_{[0,1/n]}$. Then $f_n(x)\to0$ for almost every $x$, but $\|f_n\|_p=1$. Pointwise convergence does not imply norm convergence. Conversely, norm convergence implies the existence of an almost-everywhere convergent subsequence, not necessarily pointwise convergence of the entire sequence.

In $\ell^2$, the coordinate vectors $e_n$ satisfy $e_n\rightharpoonup0$ because $\langle y,e_n\rangle=y_n\to0$ for every $y\in\ell^2$, while $\|e_n\|_2=1$. Weak convergence records every bounded linear measurement, but it can miss persistent norm.

Practical checks #

• State the measure space before claiming an $L^p$ inclusion.
• Treat functions modulo almost-everywhere equality.
• At $p=1$ and $p=\infty$, recheck duality and reflexivity instead of using the $1<p<\infty$ template.
• Use truncation to prove density only when the tail norm actually vanishes.
• Distinguish norm, weak, almost-everywhere, in-measure, and uniform convergence; each preserves different operations.