I don't know if AirPlay works with my Sony Bravia.
Need I an AppleTV Box or works it with my network?
Only if the TV supports AirPlay. If it does not say it supports AirPlay it does not support it.
My Sony Bravia TV only has an Ethernet port for the network (it's not wireless), and it doesn't support AirPlay.
Those two issues make the Apple TV my best option for using it with my iPad.
But all three together is really cool.
You need an AppleTV or download from app store DNLA apps, they are free. There is a decent one called "iMediaShare"
Another possibility is to search in the Sony TV configs, there should be an option to connect devices wirellesly.
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