Real Normed Space and Real Banach Space

In the setting of The Real Numbers: Standing Notation and Background, let $E$ be a vector space over $\mathbb{R}$, with zero vector $0_{E}$, additive inverses $-v$ and differences $u-v=u+(-v)$ as in that lemma.

1. (Norm)¶ A norm on $E$ is a map assigning to each $v\in E$ a real number $\lVert v\rVert$, subject to the following conditions for all $u,v\in E$ and every $\lambda\in\mathbb{R}$.

(a) (Positive definiteness) $0\le\lVert v\rVert$, and $\lVert v\rVert=0$ only if $v=0_{E}$.

(b) (Absolute homogeneity) $\lVert\lambda v\rVert=|\lambda|\,\lVert v\rVert$.

(c) (Triangle inequality) $\lVert u+v\rVert\le\lVert u\rVert+\lVert v\rVert$.

2. (Real normed space)¶ A real normed space is a vector space over $\mathbb{R}$ together with a norm on it. For the remainder of this item, $E$ together with $\lVert\cdot\rVert$ is a real normed space.

3. (Distance)¶ For $u,v\in E$ the distance from $u$ to $v$ is the real number $d(u,v)=\lVert u-v\rVert$. This $d$ is a metric on $E$, so that $(E,d)$ is a metric space. Indeed, $0\le d(u,v)$ by (a). If $u=v$ then $u-v=0_{E}$, and $0_{E}=0\,0_{E}$ by claim 4 of Elementary Identities in a Vector Space, so $\lVert 0_{E}\rVert=|0|\,\lVert 0_{E}\rVert=0$ by (b); conversely $d(u,v)=0$ forces $u-v=0_{E}$ by (a), whence $u=(u-v)+v=0_{E}+v=v$. Next, $(u-v)+(v-u)=0_{E}$, so $v-u$ is the additive inverse of $u-v$ by claim 2 of Elementary Identities in a Vector Space and equals $(-1)(u-v)$ by claim 5 of that lemma; hence $d(v,u)=|-1|\,\lVert u-v\rVert=d(u,v)$ by (b). Finally $u-w=(u-v)+(v-w)$ for $u,v,w\in E$, so $d(u,w)\le d(u,v)+d(v,w)$ by (c).

4. (Topological vocabulary)¶ Whenever a subset of a real normed space $E$ is called open, closed, dense or bounded, or a sequence in $E$ is called convergent or Cauchy, this refers to the metric space $(E,d)$: a subset is open if it is open in $(E,d)$, closed if it is closed in the topology these open sets form by Metric Open Sets Form a Topology, dense if it is dense there, and bounded if it is bounded in $(E,d)$; and a sequence in $E$ is convergent or Cauchy if it is convergent or Cauchy in $(E,d)$.

5. (Real Banach space)¶ $E$ is a real Banach space if the metric space $(E,d)$ is complete.