Is it possible/impossible to develop proprietary software to run on linux?

[Chris Were]

As its a legal question and a legal forum i'll flesh it out some more:

1) Is it possible/impossible to develop and distribute proprietary
software to run on linux?  The software in question is to run in user
space not kernel.
2) Is it possible to distribute this proprietary software with a
customised linux distribution, thus forming an "appliance"?
(Documentation as to the GPL OS content and linux distribution
contents, source for the GPL portions to be provided to customers, but
only binaries for the
proprietary part of the Source code)

I've been getting a lot of different opinions on this, and it seems a
fundamental question:

If we develop an application which runs on linux (and must therefore
link to some libraries for screen and keyboard, file and network IO),
must that application the be GPLv2, and must we distribute source
code?

Is it at all possible to write a proprietary application to run on linux?

I don't know much about linux at a detailed level; I come from a
Windows background (which I'd like to leave).  I am aware that the
libgcc library is licensed with an exception that appears to allow
linking of its library with other code to form a unique application
which can be distributed under any license, ie can be proprietary.
Similarly Java SE seems to have classpath exception resulting in a
similar outcome.

So does it come down to simply checking whether each library you link
to has an exception or not?  Or to careful choice of development
environment?  Or is it fullstop impossible, and am I stuck in Windows
trying to earn a livelihood?

Thanks,
Chris