In practice, the market portfolio $m$ is represented by a major market index. The measure of risk is the portfolio variance
\begin{align}
\sigma_p^2 &=\mathbf{w}^T \mathbf{V} \mathbf{w}\\
&= \sum_{i=1}^n \sum_{j=1}^n w_i w_j \sigma_{ij}\\
&= \sum_{i=1}^n w_i^2 \sigma_i^2 + \sum_{i \neq j}^{n(n-1)}w_i w_j \sigma_{ij}
\end{align}
comprised of $n$ variance terms being the diagonals of $V$ and $n(n-1)$ covariance terms being the off-diagonals of $V$. Consider an equally weighted portfolio such that $w = \frac{1}{n}$
\begin{align}
\sigma_p^2 &= \dfrac{1}{n^2} \sum_{i=1}^n \sigma_i^2 + \dfrac{1}{n^2} \sum_{i \neq j}^{n(n-1)} \sigma_{ij}
\end{align}
suppose there is an upper bound on $\sigma_i^2$ of value $L$. And an upper bound on $\sigma_{ij}$ of value $C$. We can conclude that
\begin{align}
\sigma_p^2 &\leqslant \dfrac{1}{n^2}\cdot n L + \dfrac{1}{n^2}\cdot{n(n-1)} C \\ \\
&\leqslant \dfrac{L}{n} + \bigl(1 \; – \; \dfrac{1}{n} \bigr) C
\end{align}
as $n \rightarrow \infty$,
$$\sigma_p^2 \quad \rightarrow \quad 0 \qquad + \qquad C$$
The implication is that the risk of a portfolio containing many assets is determined by the covariance contributions. Variance contributions become less significant as $n$ becomes very large. Therefore to increase the reward potential by increasing risk exposure, the covariance contributions must increase by diverifying the portfolio. Likewise, to decrease the risk exposure of the portfolio we must focus on the covariance contributions.
Consider a portfolio comprised of portfolios $p$ and $q$ with variance
\begin{align}
\sigma_p^2 &=\mathbf{w}_p^T \mathbf{V} \mathbf{w}_p\\
\sigma_q^2 &=\mathbf{w}_q^T \mathbf{V} \mathbf{w}_q\\
\end{align}respectively. What is the covriance $\sigma_{pq}$ between portfolios $p$ and $q$? It can be shown that
$$\sigma_{pq} =\mathbf{w}_q^T \mathbf{V} \mathbf{w}_p =\mathbf{w}_p^T \mathbf{V} \mathbf{w}_q$$
as covariance is symmetric.