Proposition (Properties of Orthogonal Complements)

Let $V$ be an inner product space

1. If $W\subset V$ is a Subspace, then $W^{\perp}\subset V$ is also a subspace
2. The Orthogonal Complement to the zero subspace is $$\{ 0 \}^{\perp}=V$$
3. The orthogonal complement to $V$ is $$V^{\perp}=\{ 0 \}$$
4. If $W\subset V$ is a subspace, then $W\cap W^{\perp}=\{ 0 \}$.
5. If $U,W\subset V$ are subspaces with $U\subset W$, then $W^{\perp}\subset U^{\perp}$.

Proposition (The Orthogonal complement is a complementary subspace)

If $V$ is an Inner Product Space, and let $W\subset V$ be a finite dimensional Subspace, then$$V=W\oplus W^{T}$$ i.e. their Direct Sum equals the vector space.

Proposition (Double complement returns the original subspace)

If $V$ is an Inner Product Space and $W\subset V$ is a finite dimensional Subspace then $$(W^{\perp})^{\perp}=W$$