I'm having my first linear algebra classes in college right now, and a few difficulties with the symbolism used. Missing some basics so to say.
So I have a few small questions I will just ask here:
what does the notation $C^\infty(a,b)$ as in $W = \{f \in C^\infty(a,b) \;/ \;\dfrac{d^2}{dx^2}f=0\}$ mean?
what does the notation $(u,v)\longmapsto u+v$ in the context of $V \times V \longrightarrow V$ mean?
what exactly is an 'additive unit'?
in the proof:
Prop.: If $W_1, W_2$ are two subspaces of V, then $W_1 \cap W_2$ is a subspace
(1) Since $0\in W_1$, $0\in W_2$, we have $0\in W_1 \cap W_2$.
(2) Want: If $u, v \in W_1 \cap W_2$ then $u+v \in W_1\cap W_2$.
If $u,v \in W_1 \cap W_2$ then $u,v \in W_i,\; i = 1,2$.
Since $W_i$ is a subspace we have $u,v \in W_i$.
Thus $u+v \in W_1 \cap W_2$.
(3) ...
How can he say that $W_i$ is a subspace, when that is what he is trying to prove? (or which part did i get wrong?) Also: how does the third part of the proof (if $u\in W,\alpha \in \mathbb{R}$ then $\alpha u \in W$) look, or why did my prof only write the first two?
Any help is very appreciated !
Thank you :)