20 - Properties of Annihilators

Properties of Annihilators

Let $U,W$ be subspaces of $V$ then

$$\begin{array}{rl}& (1)U\subseteq W⟹W^0\subseteq U^0\\ & (2)(U+W{)}^0=U^0\cap W^0\\ & (3)U^0+W^0\subseteq (U\cap W{)}^0\end{array}$$

Note: for $(3)$ it is equal if $\dim (V)$ is finite

Proof

$$\begin{array}{rlrl}& (1)v\in W^0& & ⟹\forall w\in W:f(w)=0\\ & & & ⟹\forall u\in U\subseteq W:f(u)=0\\ & & & ⟹f\in U^0\\ & & & \\ & (2)f\in (U+W{)}^0& & ⟺\forall u\in U,\forall w\in W:f(u+w)=0\\ & & & ⟺\forall u\in U:f(u)=0\text{ and }\forall w\in W:f(w)=0\\ & & & ⟺f\in U^0\cap W^0\\ & & & \\ & (3)f\in U^0+W^0& & ⟹\exists g\in U^0\text{ and }\exists h\in W^0:f=g+h\\ & & & ⟹\forall x\in U\cap W:f(x)=g(x)+h(x)=0\\ & & & ⟹f\in (U\cap W{)}^0\end{array}$$

For $(3)$ then it follows that $U^0+W^0\subseteq (U\cap W{)}^0$
If $V$ is finite-dimensional then

$$\begin{array}{rl}\dim (U^0& +W^0)=\dim (U^0)+\dim (W^0)-\dim (U^0\cap W^0)\\ & =\dim (U^0)+\dim (W^0)-\dim (U+W{)}^0\\ & =(\dim (V)-\dim (U))+(\dim (V)-\dim (W))-(\dim (V)-\dim (U+W{)}^0)\\ & =\dim (V)-\dim (U)-\dim (W)+(\dim (U)+\dim (W)-\dim (U\cap W))\\ & =\dim (V)-\dim (U\cap W)\\ & =\dim ((U\cap W{)}^0)\end{array}$$

Note that the first step is from Dimension Formula