Does this exist?
A client computer, same format as a notebook with screen and keyboard.
Lowpriced, low power processor, long battery time, no fan.
Something like the chromebook.
It should connect to a desktop via wifi, and be used to control the
desktop in terms of input and output.
You can do what you are suggesting with nearly any netbook and then access the desktop via remote desktop (or an equivalent tool). What, specifically, are you trying to do?
Thanks for answering.
I guess there is no market for such a device, therefor it
is not for sale.
I don't know how to explain it.
I know, I can connect 2 computers.
You have a desktop. If you want to use the desktop from another room,
then you have to extend cords for keyboard, speakers, display.
What if there was a device looking like a notebook but the only
electronics inside should be what is necessary to show output (display
and sound) and give input (keyboard).
I don't have the knowledge to build it myself.
You are describing an old "dumb terminal". There really isn't an equivalent hardware device around these days. There are some "thin clients" that could do as you propose, but since you can do in software (i.e., remote desktop) it really doesn't make much sense (or cents).
Sounds like a project, but not terribly practical.