TITLE: Why do we use $\left<\varphi,x\right>$ for $\varphi(x)$ when $\varphi$ is a linear functional?
QUESTION [4 upvotes]: 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 $V^*\times V$.

REPLY [2 votes]: 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.
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