The abstract definitions become useful only after one has a stock of model spaces. These examples also show which finite-dimensional facts survive, and which fail, in infinite dimension.
Sequence spaces
Sequence spaces give concrete laboratories where norms, duality, completeness, and compactness can be tested by coordinates.
Definition. For \(1\le p<\infty\), \(\ell^p=\{x=(x_n):\sum_n|x_n|^p<\infty\}\) with \(\|x\|_p=(\sum_n|x_n|^p)^{1/p}\). Also \(c_0=\{x:x_n\to0\}\) with \(\|x\|_\infty=\sup_n|x_n|\), and \(\ell^\infty\) is the space of bounded sequences.
Theorem. The spaces \(\ell^p\) for \(1\le p\le\infty\), \(c_0\), and \(\ell^\infty\) are Banach spaces.
Proof. For \(1\le p<\infty\), Minkowski gives the norm. If \((x^k)\) is Cauchy in \(\ell^p\), then each coordinate \(x^k_n\) is Cauchy in \(\mathbb F\), so define \(x_n=\lim_kx^k_n\). Fatou's lemma for counting measure gives \(\sum_n|x_n-x^k_n|^p\le\liminf_m\sum_n|x^m_n-x^k_n|^p\), hence \(x^k\to x\) in \(\ell^p\).
For \(\ell^\infty\), uniform limits of bounded sequences are bounded. For \(c_0\), a uniform limit of sequences tending to \(0\) also tends to \(0\): choose \(k\) with \(\|x-x^k\|_\infty<\varepsilon\), then take \(n\) large so \(|x^k_n|<\varepsilon\). Thus \(c_0\) is a closed subspace of \(\ell^\infty\).
Continuous functions
Spaces of continuous functions connect functional analysis with topology. The sup norm records uniform behavior rather than pointwise behavior.
Theorem. If \(K\) is a compact metric space, \(C(K)\) with \(\|f\|_\infty=\sup_{x\in K}|f(x)|\) is Banach.
Proof. A Cauchy sequence in \(\|\cdot\|_\infty\) is uniformly Cauchy, hence converges uniformly to a bounded function \(f\). Uniform limits of continuous functions on metric spaces are continuous: given \(x_0\), choose \(N\) with \(\|f-f_N\|_\infty<\varepsilon/3\), then use continuity of \(f_N\) near \(x_0\). Thus \(f\in C(K)\), and convergence is in the sup norm.
Lp spaces
The \(L^p\) scale is the central measure-theoretic family of Banach spaces. Its geometry changes significantly with \(p\), especially at \(p=1,2,\infty\).
Theorem. For a measure space \((\Omega,\mathcal A,\mu)\), \(L^p(\mu)\) is Banach for \(1\le p\le\infty\).
Proof. For \(1\le p<\infty\), take a Cauchy sequence and choose a subsequence \((f_{n_k})\) with \(\|f_{n_{k+1}}-f_{n_k}\|_p\le2^{-k}\). The series \(\sum_k |f_{n_{k+1}}-f_{n_k}|\) has finite \(L^p\)-norm by Minkowski, hence is finite a.e. The telescoping series therefore converges pointwise a.e. to a function \(f\). Fatou gives \(\|f-f_{n_k}\|_p\le\liminf_m\|f_{n_m}-f_{n_k}\|_p\), so the subsequence converges in \(L^p\), and Cauchyness gives convergence of the whole sequence.
For \(p=\infty\), choose representatives so that \(\|f_n-f_m\|_\infty\) controls pointwise differences outside a fixed null set for a countable cofinal set of pairs. The pointwise uniform limit outside that null set is essentially bounded and is the \(L^\infty\)-limit.
Density of simple and finite support objects
Density statements are what allow proofs to start with simple objects. Once estimates are uniform, the result extends to the completion.
Proposition. The finitely supported sequences \(c_{00}\) are dense in \(\ell^p\) for \(1\le p<\infty\) and dense in \(c_0\), but not dense in \(\ell^\infty\).
Proof. For \(x\in\ell^p\), truncate \(x^{(N)}=(x_1,\ldots,x_N,0,\ldots)\); then \(\|x-x^{(N)}\|_p^p=\sum_{n>N}|x_n|^p\to0\). For \(x\in c_0\), \(\|x-x^{(N)}\|_\infty=\sup_{n>N}|x_n|\to0\). In \(\ell^\infty\), the constant sequence \(1\) has distance \(1\) from every finitely supported sequence, so \(c_{00}\) is not dense.
Separable and nonseparable examples
Separability is a countability condition, and in analysis it often determines whether compactness can be read sequentially.
Proposition. For \(1\le p<\infty\), \(\ell^p\) is separable. The space \(\ell^\infty\) is not separable.
Proof. Sequences with finite support and rational real and imaginary parts are countable and dense in \(\ell^p\). For \(\ell^\infty\), associate to each subset \(A\subset\mathbb N\) its indicator \(1_A\). If \(A\ne B\), then \(\|1_A-1_B\|_\infty=1\). The balls \(B(1_A,1/3)\) are pairwise disjoint and uncountably many. A separable metric space cannot contain uncountably many pairwise disjoint open balls, because each such ball must contain a distinct point of a fixed countable dense set.
Vanishing at infinity
The space \(C_0\) is the locally compact analogue of continuous functions on a compact space. Vanishing at infinity supplies the compactness behavior missing from the domain.
Example. For a locally compact Hausdorff space \(S\), \(C_0(S)\) consists of continuous functions \(f\) such that for every \(\varepsilon>0\), \(\{s:|f(s)|\ge\varepsilon\}\) is compact. With the sup norm it is a Banach space: a uniform limit of continuous functions vanishing at infinity is continuous and vanishes at infinity by the same \(\varepsilon/3\) argument used for \(c_0\).