Definition

Let ${\displaystyle {\mathcal {H}}}$ be a Hilbert space and let ${\displaystyle \ell :{\mathcal {H}}\to \mathbb {C} }$ be a continuous linear transformation. Then ${\displaystyle \ell }$ is said to be a linear functional on ${\displaystyle {\mathcal {H}}}$.

The space of all linear functionals on ${\displaystyle {\mathcal {H}}}$ is denoted as ${\displaystyle {\mathcal {H}}^{*}}$. Notice that ${\displaystyle {\mathcal {H}}^{*}}$ is a normed vector space on ${\displaystyle \mathbb {C} }$ with ${\displaystyle \|\ell \|=\sup \left\{{\frac {|\ell (x)|}{\|x\|}}:x\in {\mathcal {H}};\|x\|\neq 0\right\}}$

We also have the obvious definition, ${\displaystyle \mathbf {a} ,\mathbf {b} \in {\mathcal {H}}}$ are said to be orthogonal if ${\displaystyle \mathbf {a} \cdot \mathbf {b} =0}$. We write this as ${\displaystyle \mathbf {a} \perp \mathbf {b} }$. If ${\displaystyle A\subset {\mathcal {H}}}$ then we write ${\displaystyle \mathbf {b} \perp A}$ if ${\displaystyle \mathbf {b} \perp \mathbf {a} \forall \mathbf {a} \in A}$

Theorem

Let ${\displaystyle {\mathcal {H}}}$ be a Hilbert space, let ${\displaystyle {\mathcal {M}}}$ be a closed subspace of ${\displaystyle {\mathcal {H}}}$ and let ${\displaystyle {\mathcal {M}}^{\perp }=\{x\in {\mathcal {H}}:(x\cdot a)=0\forall a\in {\mathcal {M}}\}}$. Then, every ${\displaystyle z\in {\mathcal {H}}}$ can be written ${\displaystyle z=x+y}$ where ${\displaystyle x\in {\mathcal {M}},y\in {\mathcal {M}}^{\perp }}$

Proof