Let
- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be a bounded domain
- $\mathcal T$ be a finite set of compact subsets of $\mathbb R^d$ with $$\stackrel{\circ}K_1\cap\stackrel{\circ}K_2=\emptyset\;\;\;\text{for all }K_1,K_2\in\mathcal T\tag1$$ and $$\overline\Lambda=\bigcup_{K\in\mathcal T}K\tag2$$
- $r_K\in\mathbb N$ and $\mathcal P_K\subseteq\mathbb R^K$ be a $r_K$-dimensional $\mathbb R$-vector space, for $K\in\mathcal T$
- $\mathcal N_K=\left\{N^K_1,\ldots,N^K_{r_K}\right\}$ be a basis of $\mathcal P_K^\ast$, for $K\in\mathcal T$
- $\left\{\phi^K_1,\ldots,\phi^K_{r_K}\right\}$ be the basis of $\mathcal P_K$ dual to $\mathcal N_K$, i.e. $$N^k_i(\phi^K_j)=\delta_{ij}\;\;\;\text{for all }i,j\in\left\{1,\ldots,r_K\right\}\;,\tag3$$ for $K\in\mathcal T$
Now, let $$V:=\left\{v\in C^0(\overline\Lambda):\left.v\right|_{K}\in\mathcal P_K\text{ for all }K\in\mathcal T\right\}\;.$$
Let $(\phi_i)_{i\in I}$ with $I\subseteq\mathbb N$ and $|I|<\infty$ be a basis of $V$. How are the $\phi_i$ related to the $\phi_j^K$?
In the context of finite element methods, the $\phi_i$ are called global basis functions and the $\phi_j^K$ are called local basis functions. In practice, it's clear how exactly the $\phi_j^K$ are defined by the choice of $(\mathcal P_K,\mathcal N_K)$. However, the Galerkin solution is expressed in terms of the $\phi_i$ and I haven't found how they need to be defined in any lecture note or book.
If necessary, we may assume that $\mathcal P_K$ is a space of polynomials $K\to\mathbb R$. See also my other question.
Without proper gluing conditions, there is no hope to get sensible relations between global and local basis functions.
For instance, let $\Lambda$ be an open interval split into open intervals $K$. For $K=(a,b)$ define the only local basis function $\phi^K:=x-a$. Then $\{N^K\}$ with $N^K(\phi)=\phi(b)$ is a basis of $P_K^*$. According to your definition of $V$, it holds $V=\{0\}$. So there is no global basis function at all.
I recommend reading Chapter 2 in Ciarlet's (old) book on finite element methods. There the construction of the global space from local elements is explained in detail.