Dual mappings - from U to V and from V to W

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Can someone explain to me with the help of a diagram/illustration the subject of dual mapping I'm getting confused. I know that $W$ maps using a linear function $f$, vectors to scalars. Can I write that $R$ is the $W*$? and why $S^*(f) = fS$ After all, $S^*$ is a transition from $W^*$ to $V^*$. If I run a function $f$ on it, how do I get $R$? I'm lost ... I would appreciate a verbal explanation along with an illustration Thank you!!! :) enter image description here