= Solution
With the usual variance-proxy convention for a <sub-Gaussian random vector>, every column inner product is a <sub-Gaussian random variable> with proxy at most $\tau^2$. The two-sided <Chernoff bound> and <union bound> imply
$$
\boxed{\|A^\top g\|_\infty\leq\tau\sqrt{2\log(2m/\delta)}\leq\sqrt2\tau\bigl(\sqrt{\log(2m)}+\sqrt{\log(1/\delta)}\bigr)}
$$
with <probability> at least $1-\delta$. No <independence> between these inner products is required.
\b[The printed coefficient-one bound is false under this convention.] Take $k=m=1$, $A=[1]$, and $g=\pm\tau$ with equal <probability>. This <Rademacher random variable> has the stipulated variance proxy, but for $\delta=0.99$ the printed threshold is less than $\tau$, whereas $|g|=\tau$ surely. Even a general sub-Gaussian assumption does not imply <Gaussian concentration inequality> for arbitrary Lipschitz functions. Under the stronger convention $\mathbb E e^{\langle a,g\rangle}\leq e^{\tau^2\|a\|_2^2/4}$, the same <union bound> proves the printed constants. That convention, however, differs from the binomial variance proxy used in 1(b).
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