91Ó°ÊÓ

Let: - \(V\) be a vector space over a field \(F\) - \(U\) and \(W\) be subspaces of \(V\) - \(U + W\) be the sum of the subspaces, meaning \(\{u + w : u \in U, w \in W\}\) - The annihilator of a subspace \(A\) is denoted as \(A^0\), meaning it is the set of linear functionals that map every vector in \(A\) to zero: \(A^0 = \{f \in V^* : f(a) = 0 \text{ for all } a \in A\}\), where \(V^*\) is the dual space of V (the set of linear functionals on V) The goal is to prove that: \((U+W)^{0}=U^{0} \cap W^{0}\)

Let \(f \in (U+W)^{0}\), meaning \(f(u+w) = 0\) for all \(u \in U\) and \(w \in W\). We need to show that \(f \in U^{0} \cap W^{0}\). Now, fix an arbitrary \(u \in U\). Since \(W\) is a subspace, \(0_{V} \in W\). So, for every \(u \in U\), there exists a \(w \in W\) such that \(u + w = u\). Then, by the definition of \((U+W)^{0}\): \[f(u) = f(u+ 0_{V}) = 0\] So, \(f(u) = 0\) for all \(u \in U\), which implies that \(f \in U^{0}\). Similarly, fix an arbitrary \(w \in W\). Since \(U\) is a subspace, \(0_{V} \in U\). So, for every \(w \in W\), there exists a \(u \in U\) such that \(u + w = w\). Then, by the definition of \((U+W)^{0}\): \[f(w) = f(0_V + w) = 0\] So, \(f(w) = 0\) for all \(w \in W\), which implies that \(f \in W^{0}\). Since \(f \in U^{0}\) and \(f \in W^{0}\), we conclude that \(f \in U^{0} \cap W^{0}\). Thus, \((U+W)^{0} \subseteq U^{0} \cap W^{0}\).

Let \(f \in U^{0} \cap W^{0}\), meaning \(f(u) = 0\) for all \(u \in U\) and \(f(w) = 0\) for all \(w \in W\). We need to show that \(f \in (U+W)^{0}\). Now, fix an arbitrary \(v \in U + W\). Since \(v \in U + W\), there exist \(u \in U\) and \(w \in W\) such that \(v = u + w\). Then, by the definition of \(U^{0}\) and \(W^{0}\): \[f(v) = f(u + w) = f(u) + f(w) = 0 + 0 = 0\] So, \(f(v) = 0\) for all \(v \in U + W\), which implies that \(f \in (U+W)^{0}\). Therefore, \(U^{0} \cap W^{0} \subseteq (U+W)^{0}\).

Since we've shown that \((U+W)^{0} \subseteq U^{0} \cap W^{0}\) and \(U^{0} \cap W^{0} \subseteq (U+W)^{0}\), we can conclude that \((U+W)^{0} = U^{0} \cap W^{0}\).