If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via:
\begin{align}\|A^{-1}\| & = \sup\left\{\frac{\|A^{-1}b\|}{\|b\|} : b\neq 0\right\} \\
& \qquad \left\downarrow A\text{ is a bijection }V\setminus\{0\}\rightarrow V\setminus\{0\}\right. \\
& = \sup\left\{\frac{\|A^{-1}Ab\|}{\|Ab\|} : b\neq 0\right\} \\
& = \sup\left\{\frac{\|b\|}{\|Ab\|} : b\neq 0\right\} \\
& = \frac{1}{\inf\left\{\frac{\|Ab\|}{\|b\|} : b\neq 0\right\}}\end{align}
Is there a name for the expression $\inf\left\{\frac{\|Ab\|}{\|b\|} : b\neq 0\right\}$ which is similar to the operator norm but with replacing $\sup$ with $\inf$?
Note: Above I assumed $\inf\left\{\frac{\|Ab\|}{\|b\|} : b\neq 0\right\} > 0$ which is always the case if $V$ is finite dimensional.
Best Answer
The short answer is yes. This is the smallest singular value, as a direct consequence of the min-max theorem for singular values.
For extra information you may want to consult: https://en.wikipedia.org/wiki/Singular_value (section basic properties), or for example Numerical Linear Algebra by Trefethen and Bau, see pages 94-95, if you prefer a real book.