ret More Systems and Proofs

• The distributive property is extremely useful ### 1.35 Example
• 1. If \(b = 0\) then we can divide all \(x_3\) by \(5\) and combine the last two terms to get \(F^3\), which is a vector space, without loss of generality. If not, then when you try to multiply by a scalar then you will find that the above reasoning breaks (i think).
• 1. \(f(x) = 0\) is continuous, so the additive identity exists. All sums of continuous functions result in continuous functions, so it is closed under addition. And all scalar multiples also work out.
• 1. slightly awkward: i don't actually know what a differentiable real valued function is. #todo-exr0n
• 1. (see previous)
• 1. what does it mean for a sequence of complex numbers to have a limit \(0\)? but I think you can use the same argument that the missing elements are just "collapsed" into one invisible one. ### 1.40 Definition direct sum
• Something about uniqueness?
• If there is only one way to write zero then it works (1.44 Condition for a direct sum)

I would be interested in 7, 8, 10, 12, 14-19