Fast, low weight and long battery power conflict. I've yet to see a machine that met all 3 of your requirements.
But there is another comment I want to make about running on battery power. IT'S EXPENSIVE. Most laptops come with a 300 cycle battery. If you used it everyday and charged nightly then after a year the battery might be giving about 1/2 or less the hours it was when new. Batteries run about 1 to 2 hundred so be sure you can budget that in one year.
The Acer Timelinex models are worth a look for long battery times.
Which laptop should I get?
Most important to me is long battery, life VERY fast machine, low weight. I am using this primarily for school and business functions - primarily Internet and Microsoft Office Suite (Excel, Word, Powerpoint).
I have been told that ASUS is the best here but some have suggested Sony,,,,,?