In this post, the trace is given as an example of dual space for the vector space composed of $2 \times 2$ real-valued matrices. Coincidentally, just today I ran across the same example for the inner product in this youtube video.
So the dual space $V^*$ is the set of linear maps or linear functionals from $V$ to the real numbers (field), $V^*:V\to \mathbb R.$ As maps, the dual space is a homomorphism, $\text{Hom}(V,\mathbb R)$, itself forming a vector space equipped with addition and scalar multiplication:
$(\varphi+\psi)(x)= \varphi(x)+\psi(x)$
$(\alpha\varphi)(x)=\alpha(\varphi(x))$
Similarly an inner product space is a pair of a vector space, $V$, paired with a function $\langle\cdot,\cdot\rangle$ from $V\times V\to \mathbb R$, fulfilling:
(i) $\langle v, w\rangle=\langle v, w\rangle$
(ii) $\langle v+y, w\rangle=\langle v, w\rangle+\langle y, w\rangle$
(iii) $\langle c\,v, w\rangle=c\langle v, w\rangle$
(iv) $\langle v, v\rangle\geq0$
Extremely similar, parallel concepts, including both vector spaces, maps to the underlying field elements ($\mathbb R)$, and, clearly, the "exotic" example of the trace of matrices.
So where do these concepts start to differ? And why are they so similar in so many ways?
Best Answer
They are different things by definitions. One might say that they look similar because both of them have "linearity" as a key ingredient in the definitions: for dual spaces, one has linear forms; for inner product spaces, one has bilinear forms.
A well known connection between these two concepts is given by the Riesz representation theorem.
Let $V$ be a real Hilbert space (namely, complete inner product space over $\mathbb{R}$). On the one hand, for any fixed $x\in V$, $$ y\mapsto \langle x,y\rangle\tag{1} $$ gives you a (continuous) linear functional on $V$, namely an element in $V^*$. On the other hand, the Riesz representation theorem says that every element in $V^*$ (assuming $V^*$ means the continuous dual) are of the form (1).