Orthogonal Projection in a Hilbert Space

• categories: Functional analysis, Real Analysis, Definition

In a Hilbert Space $H$, the orthogonal projection onto a closed subspace $M\subset H$ is the operation that maps any vector in $H$ to its “closest” vector in $M$. This projection has key properties and is a foundational concept in functional analysis and applications such as Fourier analysis.

Definition

Let $H$ be a Hilbert space, and let $M\subset H$ be a closed subspace. For any vector $v\in H$, there exists a unique vector $u\in M$ such that:

$$\Vert v-u\Vert =\underset{w\in M}{\inf }\Vert v-w\Vert .$$

The vector $u$ is called the orthogonal projection of $v$ onto $M$, and it is denoted by $P_M(v)$.

Properties

1. Existence and Uniqueness: For any $v\in H$, there exists a unique projection $u=P_M(v)$ in $M$ such that $v-u$ is orthogonal to every vector in $M$. That is,

   $$⟨v-P_M(v),w⟩=0\text{for all }w\in M.$$

2. Idempotence: The projection operator $P_M$ satisfies $P_M^2=P_M$, meaning that projecting twice has the same effect as projecting once.

3. Linearity: $P_M$ is a Linear map on $H$.

4. Norm Non-Increasing: For any $v\in H$, we have $\Vert P_M(v)\Vert \le \Vert v\Vert$.

5. Orthogonality Decomposition: Every vector $v\in H$ can be uniquely decomposed as:

   $$v=P_M(v)+(v-P_M(v)),$$

   where $P_M(v)\in M$ and $(v-P_M(v))\in M^{\perp }$ (the orthogonal complement of $M$ in $H$).

Geometric Intuition

The orthogonal projection $P_M(v)$ is the “shadow” of $v$ onto the subspace $M$, meaning it is the vector in $M$ that minimizes the distance to $v$. This concept is similar to projecting a point onto a line in Euclidean space, where the projection is the point on the line closest to the original point.

Example in $L^2$ Space

In the Hilbert space $L^2([a,b])$, let $M$ be the subspace of functions spanned by a set $\{e_1,e_2,\dots ,e_n\}$ of orthonormal functions. The orthogonal projection of $f\in L^2([a,b])$ onto $M$ is:

$$P_M(f)=\sum _{i=1}^n⟨f,e_i⟩e_i.$$