User:IssaRice/Linear algebra/Riesz representation theorem

Let's take the case where $V=\mathbf {R} ^{n}$ and the inner product is the usual dot product. What does the Riesz representation theorem say in this case? It says that if we have a linear functional $T:\mathbf {R} ^{n}\to \mathbf {R}$ then we can write $T$ as $Tv=v\cdot u$ for some vector $u\in \mathbf {R} ^{n}$. But we already know (from the correspondence between matrices and linear transformations) that we can represent $T$ as a 1-by-n matrix. And v can be thought of as a n-by-1 matrix. And now the dot product is the same thing as the matrix multiplication!

If $\sigma =(e_{1},\ldots ,e_{n})$ is the standard basis of $\mathbf {R} ^{n}$ and $(1)$ is the standard basis of $\mathbf {R}$, then $Tv=[Tv]^{(1)}=[T]_{\sigma }^{(1)}[v]^{\sigma }=[T]_{\sigma }^{(1)}\cdot [v]^{\sigma }=[v]^{\sigma }\cdot [T]_{\sigma }^{(1)}=v\cdot [T]_{\sigma }^{(1)}$.

So in the case $V=\mathbf {R} ^{n}$ we can understand the Riesz representation theorem as saying something we already knew. What Riesz representation theorem does is extend this same sort of "representability" to all finite-dimensional V and all linear functionals $T:V\to \mathbf {F}$.