Spanning Sets
\(\text{span}\{|v_1\rangle,\cdots,|v_2\rangle\} = \{|v\rangle : |v\rangle=\sum_i^n c_i|v_i\rangle,c_i\in\mathbb{F}\}\) = {all possible linear combinations of the \(|v_i\rangle\) }
Definition
The set of linear combinations of \(|v_1\rangle, |v_2\rangle, \cdots, |v_n\rangle\) is called the span of \(|v_1\rangle, |v_2\rangle, \cdots, |v_n\rangle\). The span of \(|v_1\rangle, |v_2\rangle, \cdots, |v_n\rangle\) is denoted by \(\text{span}{|v_1\rangle, |v_2\rangle, \cdots, |v_n\rangle}\).
Examples
Unbounded spaces: span{line segment} = line. span{two non-collinear line segments} = plane. Continuous Function Spaces.
Bounded spaces: Integer rings.