1.1.4

An agent is just something that acts (agent comes from the Latin agere, to do).

Actions include: operate autonomously, perceive their environment, persist over a prolonged time period, adapt to change, and create and pursue goals.

A rational agent is one that acts so as to achieve the best outcome or, when there is uncertainty, the best expected outcome.

state space 

Thinking about a graph, a state = vertex. The state space refers to all the possible states, so as you say, it is equivalent to all the possible vertexes in the graph.

When we say that the tuple \langle s_0, A , f \rangle⟨s0​,A,f⟩, where s_0s0​ is the initial state, AA the actions, and ff the transition function, defines implicitly the state space, it refers to all the vertexes. Now, the same tuple also describes the edges (transition from one state to another), which can be used to define the search space.

The search space talks about the graph (tree) of states and actions (vertexes and edges), that a search algorithm has to traverse/explore to find a solution.(Nir Lipovetzky, Prof, University of Melbourne)