I have looked through books such as Matrix Analysis by R.A. Horn and C.R. Johnson and would not find an answer to the following question:


Given $V^TV \in S^{n}$, where $V$ is an invertible matrix with each column of $V$ of unit length. Can the norm of $(V^TV)^{-1}$ be bounded above by a constant that does not depend on the given matrix $V$? If not, please provide an upper bound in terms of $V$?



An upper bound on an invertible matrix
As noted in the comments, the quantity you want is $\sigma\_{\min}(V)^{-2}$, the inverse square of the minimum singular value of $V$.


Unfortunately you can't get any meaningful bound from below for matrices with unit column norms, as they can still be arbitrarily close to singular. For instance,
$$
\begin{bmatrix}
1 & \cos \alpha \\
0 & \sin \alpha \\
0 & 0
\end{bmatrix}
$$
has a singular value that must tend to $0$ when $\alpha \to 0$.


