Metric Space VS Normed Space VS Inner Product Space I am taking my first course in topology and the lecturer briefly mentioned those $3$ spaces: 1.Metric Space 2. Normed Space 3. Inner Product Space 
For better understanding, I was wonder, how are those 3 connected? for example I know that $||x||=\sqrt{<x,x>}$ so the norm is a "subset" or there are inner products that induce a norm? and a metric space is a "subset" of norm space?
 A: The three types of spaces imply increasingly greater assumptions of "structure": 


*

*In a metric space we we can talk about the distance between points, $d(x,y)$, which allows for topological concepts of open sets, etc. 

*In a normed space we can in addition talk about the size of a given point: $||x||$. Any norm $||\cdot||$ induces a metric: $d(x,y) = ||x - y||$. 

*In an inner product space we can in addition talk about the "angles" between points, and define projections (and all nice things that we do with inner products). Any inner product $(\cdot,\cdot)$ induces a norm: $||x|| = \sqrt{(x,x)}$.
A: Every inner product induces a norm, but not vice versa.
Every norm induces a metric, but not vice versa.
Thus you have something like $$ \left\{ \mathrm{Inner\ Product\ Spaces} \right\} \subsetneq \left\{ \mathrm{Normed\ Spaces} \right\} \subsetneq \left\{ \mathrm{Metric\ Spaces} \right\}$$
A: You have $\text{IPS} \subsetneq \text{NS} \subsetneq{MS}$. 
To conclude that $\text{IPS} \subset \text{NS}$, as you write above, consider the norm $\|x\| = \sqrt{\langle x, x \rangle}$. To conclude that $\text{NS} \not\subset \text{IPS}$, consider $L^p$ for $p \neq 2$ (or any non-reflexive Banach space). 
To conclude that $\text{NS} \subset \text{MS}$, consider the metric $d(x,y) = \|x - y\|$. To conclude that $\text{MS} \not\subset \text{NS}$, use any metric space which is not a vector space (metric spaces don't need to be vector spaces, but normed spaces --and inner-product spaces-- do).
A: Examples in $2$-dimensional space $\mathbb R^2$:
$\bullet\quad$ $l^2$ norm: $\|(x,y)\| = (x^2+y^2)^{1/2}$  .  This space is an inner product space, a normed space, and a metric space.  
$\bullet\quad$ $l^p$ norm, $1 \le p < \infty, p \ne 2$: $\|(x,y)\| = (x^p+y^p)^{1/p}$  .  This space is a normed space, and a metric space, but not an inner product space.  
$\bullet\quad$ $l^p$ distance, $0 < p < 1$: $d((x_1, y_1), (x_2, y_2)) = (x_1-x_2)^p+(y_1-y_2)^p$  .  This space is a metric space, but not an inner product space and not a normed space.  
