Nearest-Point Projection onto a Nonempty Closed Convex Subset of the Lebesgue Space of Square-Integrable Vector-Valued Functions

Let $T>0$ and $d$ be as in the definition of the Lebesgue space $L^{2}([0,T];\mathbb{R}^{d})$, write $H=L^{2}([0,T];\mathbb{R}^{d})$, and adopt the pairing $\langle\cdot,\cdot\rangle_{L^{2}}$, the norm $\lVert\cdot\rVert_{L^{2}}$ and the metric $d_{L^{2}}$ of that definition.

Let $C$ be a nonempty subset of $H$ that is convex and closed for the topology of open subsets determined by $d_{L^{2}}$. Then the following hold.

1. (Existence and uniqueness.) For every $u\in H$ there is exactly one element of $C$, written $\pi_{C}(u)$, such that

$$\lVert u-\pi_{C}(u)\rVert_{L^{2}}\le\lVert u-v\rVert_{L^{2}}\qquad\text{for every }v\in C .$$

The map $\pi_{C}:H\to C$ so defined is called the nearest-point projection onto $C$.

2. (Variational inequality.) For $u\in H$ and $p\in C$, the equality $p=\pi_{C}(u)$ holds if and only if

$$\langle u-p,\;v-p\rangle_{L^{2}}\le0\qquad\text{for every }v\in C .$$

3. (Points of $C$ are fixed.) $\pi_{C}(u)=u$ for every $u\in C$; in particular $\pi_{C}$ maps $H$ onto $C$.

4. (Nonexpansiveness.) For all $u,u'\in H$ one has $\lVert\pi_{C}(u)-\pi_{C}(u')\rVert_{L^{2}}\le\lVert u-u'\rVert_{L^{2}}$; that is, $\pi_{C}$ is Lipschitz with constant $1$.