I am wondering what exactly is the relationship between the three aforementioned spaced. All of them seem to show up many times in: Linear Algebra, Topology, and Analysis. However, I feel like I'm missing the bigger picture of how these spaces relate to each other. For example, in my course in multi-dimensional analysis, we started out talking about metric spaces, but later suddenly switched to normed vector spaces, without any explicit mention of this transition. In linear algebra we usually talked about inner product spaces, and in topology we talked about metric spaces and topological spaces.
The bigger picture of the relation between these three is still unclear to me. Which is used where, for what reason, and how do they relate?
I do know the definitions of all three of them:
A metric space is a pair $(S,d)$ with $S$ a set and $d: S \times S \to \mathbb{R}_{\geq 0}$ a metric:
- $d(x,x) = 0$ for all $x \in S$ and $d(x,y) >0$ for $x \neq y$,
- $d(x,y) = d(y,x)$,
- $d(x,z) \leq d(x,y) + d(y,z)$.
A (real) inner product space is a pair $(V,\langle \cdot \rangle)$ where $V$ is a (real) vector space and $\langle \cdot \rangle: V \times V \to \mathbb{R}$ is an inner product:
- $\langle v,w \rangle = \langle w,v \rangle$,
- $\langle a_1 v_1 + a_2v_2,w \rangle = a_1\langle v_1,w \rangle + a_2\langle v_2,w \rangle$ for all $a_1,a_2 \in \mathbb{R}$,
- $v \neq 0 \Longrightarrow \langle v,v \rangle > 0$.
A (real) normed vector space is a pair $(V,\|\cdot\|)$ where $V$ is a (real) vector space and $\|\cdot\|: V \to \mathbb{R}_{\geq 0}: v \mapsto \|v\|$ is a norm on $V$:
- $\|v\| \geq 0$ and $\|v\| = 0 \ \Longleftrightarrow \ v = 0$.
- For $t \in \mathbb{R}$ and $v \in V$ we have $\|tv\| = |t|\|v\|$
- $\|v+w\| \leq \|v\| + \|w\|$.
I also know that an inner product gives rise to a norm by taking $\|v\| = \sqrt{\langle v,v \rangle}$, for example the Euclidean norm derives from the standard inner product on $\mathbb{R}^n$ in this way. And Cauchy-Schwarz: $|\langle x,y \rangle| \leq \|x\|\|y\|$.
I'm not interested in details about the definitions but in the intuition and bigger picture of these three spaces, and how they show up in Analysis.