Let us say that a normed vector space has the
a) RR (Radon-Riesz) property if for any sequence, norm convergence is equivalent to weak convergence + convergence of norms.
b) KK (Kadec-Klee) property if the weak topology coincides with the norm topology on the unit sphere.
Observations:
o) Property b) implies property a).
o) A locally uniformly convex space has a).
Questions:
i) Is the terminology appropriate?
ii) Are the two properties equivalent?
iii) If not ii), give a counterexample.
Example:
The sequence space $\ell_1$ has property a); in fact it has the stronger Schur property. It is also has property b): Since the norm topology is finer than the weak topology, we only need to show that if $x$ is in the restriction $P := U \cap S$ of a norm open set $U$ to the unit sphere $S$, then $x$ is contained in the restriction $Q := W \cap S$ of a weakly open set $W$ to $S$, and $Q \subset P$. Wlog, $U$ is a small ball of radius $\epsilon > 0$, so that $P = \{ y \in S : |x - y|_1 < \epsilon \}$, and wlog $Q = \{ y \in S : \forall f \in F : |f(x-y)| < \delta \}$, where $F \subset \ell_\infty$ is finite and $\delta > 0$. Now we can partition the indices $x = a + b$ (into a finite and an infinite part) such that $|b|_1 < \epsilon/4$, with the same partition for any $y = p + q$. Then $|x - y|_1 > \epsilon$ implies $|a - p|_1 \geq \epsilon / 4$. So take $\delta := \epsilon / 4$, and $F$ as the sequences $f$ that have all combinations of $\pm1$ on the support of $a$ and zero outside. Then $|a - p|_1 \geq \epsilon / 4$ implies $|f(x - y)| \geq \delta$ for some $f \in F$. Hence $y \not\in P$ implies $y \not\in Q$, or $Q \subset P$.
The two properties are equivalent as long as you only consider sequences:
$RR\implies KK$ because on the unit sphere, by definition, all norms are 1, therefore the norm obviously converges. For the other direction, just scale your sequences, then you get $$ \frac{f_n}{\|f_n\|}\rightarrow \frac{f}{\|f\|} $$ if the norms converge, this converges strongly according to KK.
However, the weak topology is not first-countable in general, therefore it cannot be fully characterized by sequences.
For the KK property, your terminology therefore is not fully correct, the KK property says that sequential(!) weak convergence on the unit ball is equivalent to the norm convergence. The topologies are for sure not equivalent, as the weak topology is not first-countable. Thanks to Nate for this correction, I overlooked this.