The Trace Linear Functional on X*⊗X

# The Trace Linear Functional on X*⊗X

Consider the bilinear functional from $T: X^* \times X \to \mathbb{F}$ defined for all $f \in X^*$ and all $x \in X$ by:

(1)
\begin{align} \quad T(f, x) = f(x) \end{align}

Since $T : X^* \times X \to \mathbb{F}$ is bilinear there exists a unique linear map $\tilde{T} : X^* \otimes X \to \mathbb{F}$ such that:

(2)
\begin{align} \quad \tilde{T}(f \otimes x) = f(x) \end{align}

We will denote the map $\tilde{T}$ above by $\mathrm{tr}$ and give it a special name.

 Definition: Let $X$ be a normed linear space. The Trace of $X$ is the unique linear functional $\mathrm{tr} : X^* \otimes X \to \mathbb{F}$ such that $\mathrm{tr}(f \otimes x) = f(x)$ for all $f \in X^*$ and for all $x \in X$.

Let $u = \sum_{i=1}^{n} f_i \otimes x_i \in X^* \otimes X$. Then:

(3)
\begin{align} \quad \mathrm{tr}(u) = \sum_{i=1}^{n} \mathrm{tr}(f_i \otimes x_i) = \sum_{i=1}^{n} f_i(x_i) \end{align}