\subsection{Exploiting Low Dimensional Structure}
In this section, we consider the low-rank structure in a linear functional $\Lb$. Recall that $\Lb$ is an r-dimensional linear functional. This provides us to represent the linear functional $\Lb$ over any feature point $\phi$ with an r-dimensional structure. Using Riesz's Representation Theorem, we can write $\Lb\phi$ for any $\phi$ as follows:
\begin{eqnarray*}   \Lb\phi=\sum_{k=1}^\infty\langle \phi, \tau_k\rangle \mathbf{e}_k
\end{eqnarray*}
As $\Lb$ is low rank with rank r, we can write each $\tau_k$ as $\tau_k=\sum_{j=1}^rv_{k,j}\beta_j$, where $\{\beta_1, \ldots, \beta_r\}$ is an orthonormal basis for the range of linear functional $\Lb$ and $v_{k,j}\in \mathbb{R}, \forall k,j$. Therefore,
\begin{eqnarray*}
    \langle \Lb\phi, \Lb\phi'\rangle &=& \sum_{k=1}^\infty\langle \phi, \tau_k \rangle \langle \phi', \tau_k \rangle \\ &=& \sum_{i=1}^r\sum_{j=1}^r \left(\sum_{k=1}^\infty v_{k,j}v_{k,i}\right)\langle \phi, \beta_i \rangle \langle \phi', \beta_j \rangle \\
    &=& \vartheta^T\mathbf{G} \vartheta' \label{Riesz_representation}
\end{eqnarray*}
where $\Gb_{i,j}=\sum_{k=1}^\infty v_{k,j}v_{k,i}$. We can expand the $\Lb: \Vb\mathcal{B}^T$ for the linear map $\Vb$ from $\mathbb{R}^r$ to $\mathcal{H}$.
Consider $\Gb=\Vb^T\Vb$, where $\Vb\in \mathbb{R}^{\infty \times r}$ and each $\tau_k = \beta\Vb_{(k,:)}$, $\tau = \beta \Vb^T$. 

We consider the subspace $\mathcal{S}_\mathcal{X}\subset\mathcal{H}$ spanned by $\{\phi(\xb_1), \phi(\xb_2), \ldots \phi(\xb_n)\}$. Recall that $\psi(\xb_1), \psi(\xb_2), \ldots \psi(\xb_n)$ be the n principal component directions in this space, $\Lb$ can be represented as $\Ub\Ab^T\Phi^T$. Similarly, as $\Lb$ is an r-dimensional linear functional, we can write it as $\Lb=\Vb\mathcal{B}^T$. Therefore, we have that $\Ub\Ab^T\Phi^T$ and $\Vb\mathcal{B}^T$ are the same operators over the subspace $\mathcal{S}_\mathcal{X}$ and we can write
\begin{eqnarray*}
    \Ub\Ab^T\Phi^T=\Vb\overline{\mathcal{B}}^T
\end{eqnarray*}
where $\overline{\mathcal{B}}=\Psi(\Psi^T\Psi)^{-1}\Psi^T\mathcal{B}=\Psi\Psi^T\mathcal{B}$
\begin{remark}
If one assumes that data is well spread out and it excites almost all the energy in r-dimensional structure of $\Lb$, then we can suppose that $\|\mathcal{B}-\Psi\Psi^T\mathcal{B}\|_F^2<\epsilon$. In the ideal case, we have $\Psi\Psi^T\mathcal{B}=\mathcal{B}$.
\end{remark}
 
\begin{eqnarray*}
    \|\Phi\Ab\Mb\Ab^T\Phi^T\|_F^2=\|\mathcal{B}\Gb\mathcal{B}^T\|_F^2
\end{eqnarray*}


\begin{eqnarray*}
    \Ub\Ab^T\Phi^T=\Vb\mathcal{B}^T\Psi\Psi^T \\
    \Ub\Ab^T\Kb=\Vb\mathcal{B}^T\Psi\Psi^T\Phi=\Vb\mathcal{B}^T\Psi\Ab^T\Kb
\end{eqnarray*}

\begin{eqnarray*}
    \Ub\Ab^T\Phi^T=\Vb\mathcal{B}^T\Psi\Psi^T \\
    \Ub=\Vb\mathcal{B}^T\Psi
\end{eqnarray*}



Assuming that data is well spread out so that it can excite all the energy in r-dimensional structure of $\Lb$, we can claim that

Then, we find that $\Ub=\Vb\mathcal{B}^T\Psi$, which gives $\|\Mb\|_F^2=\|\Gb\|_F^2$ kk $\Ub^T\Ub=\Mb=\Psi^T\mathcal{B}\Gb\mathcal{B}^T\Psi$. Therefore, we have 
\begin{eqnarray*}
    kk
\end{eqnarray*}




\begin{corollary}
    kkk
\end{corollary}
\textbf{Proof:}
From the fact that $k$
To have a more clear understanding, we provide a short discussion below. 


Then, We  note that can write the inner product of $\Lb\phi_i$ and $\Lb\phi_j$ as follows:
\begin{eqnarray*}
    \langle\Lb\phi_i,\Lb\phi_j\rangle_{\mathcal{H}}=\varphi_i^T\Mb\varphi_j.
\end{eqnarray*}
 As $\Lb$ is an r-dimensional linear functional, we can also write the inner product of $\Lb\phi_i$ and $\Lb\phi_j$ as follows:
\begin{eqnarray*}
       \langle\Lb\phi_i,\Lb\phi_j\rangle_{\mathcal{H}}=\psi_i^T\mathbf{G} \psi_j 
\end{eqnarray*} 