Normed and Banach Spaces - Key Proof

We present a detailed proof of the completion theorem, showing that every normed space can be embedded isometrically into a Banach space as a dense subspace.

TheoremCompletion of Normed Spaces

Let $(X, \|\cdot\|)$ be a normed space. Then there exists a Banach space $(\widehat{X}, \|\cdot\|)$ and an isometric linear embedding $\iota: X \to \widehat{X}$ such that $\iota(X)$ is dense in $\widehat{X}$ . Moreover, this completion is unique up to isometric isomorphism.

Proof

Construction: Let $\mathcal{C}$ be the set of all Cauchy sequences in $X$ . Define an equivalence relation on $\mathcal{C}$ by $(x_n) \sim (y_n) \iff \lim_{n \to \infty} \|x_n - y_n\| = 0$

Let $\widehat{X} = \mathcal{C}/\sim$ be the quotient space. For a Cauchy sequence $(x_n)$ , denote its equivalence class by $[(x_n)]$ .

Vector Space Structure: Define operations on $\widehat{X}$ by $[(x_n)] + [(y_n)] = [(x_n + y_n)], \quad \alpha [(x_n)] = [(\alpha x_n)]$

These are well-defined (independent of representatives) and make $\widehat{X}$ a vector space over $\mathbb{K}$ .

Norm: For a Cauchy sequence $(x_n)$ , the sequence $(\|x_n\|)$ is Cauchy in $\mathbb{R}$ , hence convergent. Define $\|[(x_n)]\| = \lim_{n \to \infty} \|x_n\|$

This is well-defined and defines a norm on $\widehat{X}$ .

Embedding: Define $\iota: X \to \widehat{X}$ by $\iota(x) = [(x, x, x, \ldots)]$ (the constant sequence). Then:

$\iota$ is linear: $\iota(\alpha x + \beta y) = \alpha \iota(x) + \beta \iota(y)$
$\iota$ is isometric: $\|\iota(x)\| = \lim_{n \to \infty} \|x\| = \|x\|$
$\iota(X)$ is dense: Every $[(x_n)] \in \widehat{X}$ is the limit of $\iota(x_n)$

Completeness: Let $(z_k)$ be a Cauchy sequence in $\widehat{X}$ where $z_k = [(x_n^{(k)})]$ . For each $k$ , choose $x_k \in X$ such that $\|\iota(x_k) - z_k\| < 1/k$ . Then $(x_k)$ is Cauchy in $X$ , so $[(x_k)] \in \widehat{X}$ is the limit of $(z_k)$ .

Uniqueness: If $(\widehat{X}_1, \iota_1)$ and $(\widehat{X}_2, \iota_2)$ are two completions, the denseness and continuity force a unique isometric isomorphism $\Phi: \widehat{X}_1 \to \widehat{X}_2$ with $\Phi \circ \iota_1 = \iota_2$ .

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ExampleConcrete Completions

Polynomials to Continuous Functions: The space of polynomials $\mathbb{R}[x]$ on $[0,1]$ with supremum norm completes to $C[0,1]$
Rational Numbers: The normed space $(\mathbb{Q}, |\cdot|)$ completes to $(\mathbb{R}, |\cdot|)$
Step Functions: Simple functions on a measure space complete to $L^p$ spaces

Remark

The completion theorem is a powerful existence result that allows us to work in complete spaces even when starting from incomplete ones. In practice, many important function spaces arise naturally as completions of simpler spaces.

This construction parallels the completion of metric spaces but preserves the additional linear and norm structures, making it a cornerstone of functional analysis.