existence and uniqueness of solution of ordinary differential equations

Let $E\subset W$ where $E$ is an open subset of $W$ which is a normed vector space, and let $f$ be a continuous differentiable map

$$f:E\to W.$$

Then the ordinary differential equation defined as

$$\dot{x}=f(x)$$

with the initial condition

$$x(0)=x_0$$

where $x_0\in E$ has a unique solution on some interval containing zero. More specifically there exists $\alpha >0$ such that the following is a unique function

$$x:(-\alpha ,\alpha )\to E$$

such that $\dot{x}=f\circ x$ and $x(0)=x_0$.[HS]