(S. T. Dilip, Chennai)

Both sentences are correct. You can use `half' or `half of' before a `noun with a determiner (article, possessive or demonstrative)'. For example, you can say, `I finished half of my work by 6:00 pm'. You can also say, `I finished half my work by 6:00 pm.'

Half (of) her adult life has been spent in jail.

Before pronouns, however, we use only `half of' and not `half'. For example, `These chocolates are really good. I've already eaten half of them.' We cannot say, `I've already eaten half them'. When we talk about measurements or quantity, then we use `half' and not `half of'. There's only half a bottle of shampoo left.