Riesz-Frechet Representation of Bounded Linear Functionals on 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 $\langle\cdot,\cdot\rangle_{L^{2}}$ and the norm $\lVert\cdot\rVert_{L^{2}}$ of that definition.

Let $\Lambda:H\to\mathbb{R}$ be a map and let $C$ be a real number with $C\ge0$ such that

(i) (Linearity.) $\Lambda$ is linear from the real vector space $H$ to $\mathbb{R}$, that is, $\Lambda(su+s'v)=s\,\Lambda(u)+s'\,\Lambda(v)$ for all $u,v\in H$ and all real $s,s'$;

(ii) (Boundedness.) $|\Lambda(u)|\le C\,\lVert u\rVert_{L^{2}}$ for every $u\in H$, the absolute value being that of $\mathbb{R}$.

Then there is exactly one $\eta\in H$ such that

$$\Lambda(u)=\langle\eta,u\rangle_{L^{2}}\qquad\text{for every }u\in H,$$

and this $\eta$ satisfies $\lVert\eta\rVert_{L^{2}}\le C$.