Why do we use $\left<\varphi,x\right>$ for $\varphi(x)$ when $\varphi$ is a linear functional? Let $V$ a $K-$vector space and $V^*$ it's dual. What is the motivation of using the notation $\left<\varphi,x\right>$ for $\varphi(x)$ ? Is it a consequence of the fact that \begin{align*}
\left<\cdot ,\cdot \right>: V^*\times V&\longrightarrow K\\ (\varphi,x)&\longmapsto \left<\varphi,x\right>=:\varphi(x)
\end{align*}
would be a scalar product ? But it looks strange since a scalar product must take element form $V\times V$ or $V^*\times V^*$, but not of the form $V^*\times V$. 
 A: The fact that any linear functional can be expressed as an inner product is true only if the vector space $V$ is a Hilbert space.
This is the content of the Riesz Representation Theorem.
In this case a linear functional $\varphi(y)$ can be identified with the inner product of a vector $x$ with $y$ so that we can write $\varphi(y)=\langle x,y\rangle$. The functional $\varphi$  and the vector $x$ are dual and , with an abuse of notation, we use the same symbol to indicate both them, so we can write $\varphi(y)=\langle \varphi, y\rangle$, but really the second $\varphi$ is an element of the hilbert space  and the first is the the corresponding element (by Riesz Theorem) in the dual space.
A: For any linear functional $\varphi$, there exists a vector $v$ such that $\varphi(x) = \langle v,x \rangle$ for all $x$. This is actually a one-to-one correspondence between $V^{*}$ and $V$. Thus it makes practical sense to represent a linear functional $\varphi$ as an element of $V$, with the convention that $\varphi(x) = \langle \varphi,x \rangle$.
A: When $V $  is a Hilbert space (a very common situation, for instance in physics), you have  $V^*=V $, so it is an actual inner product.
A: As Lee Mosher wrote in a comment, the brackets $\langle \cdot ,\cdot \rangle$ are very common for bilinear maps.
These are maps $V\times W \to \mathbb{R}$, where $V$ and $W$ are real vector spaces, such that
$$
\langle ax+by,z\rangle = a\langle x,z\rangle + b\langle y,z\rangle \quad\text{for $a,b\in\mathbb{R}$ and $x,y\in V$ and $z\in W$} \\
\langle x,ay+bz\rangle = a\langle x,y\rangle + b\langle x,z\rangle \quad\text{for $a,b\in\mathbb{R}$ and $x\in V$ and $y,z\in W$.} $$
The notation for linear functionals is a special case where $W = V^*$.
(I wrote it here for vector spaces over the real numbers, but this works equally well for vector spaces over other fields, or even modules over rings.)
As to why this notation is used, I can't be sure. However, from experience I can say that it is often convenient to have a notation that looks more symmetric than function application, when you have two spaces that are dual to each other, but it is not clear which one should be $V$ and which one should be $V^*$ (after all, $(V^*)^*$ is just $V$ -- at least for sufficently nice spaces).
Note: In quantum mechanics, there is a very similar notation called Bra-ket notation. If you want to name your quantum state $\psi$, you write it $\mid \psi \rangle$. If you want to name an element of the dual space $\phi$, you write it $\langle \phi \mid$. The evaluation of $\phi$ as $\psi$ is then written $\langle \phi \mid \psi\rangle$. It was introduced by Paul Dirac in 1939 in A new notation for quantum mechanics.
