Update 4_vector_spaces_QM.md

src/4_vector_spaces_QM.md

@@ -127,10 +127,10 @@ $$
 \hat{I} = \sum_i |\phi_i\rangle|\phi_i\rangle \, ,
 $$
 We can insert this  identify operator within the bracket to evaluate the inner
-product $\langle \chi|\psi\rangle$ between the two state vectors to evaluate the inner product $\langle \chi}|\psi\rangle$:
+product $\langle \chi|\psi\rangle$ between the two state vectors to evaluate the inner product $\langle \chi|\psi\rangle$:
 $$
 \langle \chi|\psi\rangle=
-\langle\chi|\hat{I} |\psi\rangle=\sum_i\langle\chi|\phi_i\rangle\langle\phi_n}|\psi\rangle \, .
+\langle\chi|\hat{I} |\psi\rangle=\sum_{i=1}^n \langle\chi| \phi_i \rangle \langle\phi_i|\psi\rangle \, .
 $$
 Next, using that $\chi_n = \langle \phi_i|\chi \rangle$ are the components of the
 state vector $|\chi\rangle$ and that  $\langle \chi| \phi_n \rangle=(\langle\phi_n|\chi\rangle)^*$,