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 $\sigma =(e_1,\dots ,e_n)$ 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 inner product spaces V and all linear functionals $T:V\to \mathbf{F}$.

this video talks about this for ${\mathbf{R}}^2$