Subspace A subset U⊆VU\subseteq VU⊆V is a subspace if it satisfies: contains zero vector closed under addition ∣u⟩,∣v⟩∈U,∣u⟩+∣v⟩∈U\left\lvert u \right\rangle,\left\lvert v \right\rangle\in U, \left\lvert u \right\rangle+\left\lvert v \right\rangle\in U∣u⟩,∣v⟩∈U,∣u⟩+∣v⟩∈U closed under scalar multiplication ∣u⟩∈U,c∈F⇒c∣u⟩∈U\left\lvert u \right\rangle\in U, c\in \mathbb{F}\Rightarrow c\left\lvert u \right\rangle\in U∣u⟩∈U,c∈F⇒c∣u⟩∈U Trace