Properties of Elementary Tensors of Normed Linear Spaces

Proposition 1: Let $X$ and $Y$ be normed linear spaces. Then: a) $(x_1 + x_2) \otimes y = x_1 \otimes y + x_2 \otimes y$ for all $x_1, x_2 \in X$ and for all $y \in Y$. b) $x \otimes (y_1 + y_2) = x \otimes y_1 + x \otimes y_2$ for all $x \in X$ and for all $y_1, y_2 \in Y$. c) $(\alpha x) \otimes y = \alpha (x \otimes y) = x \otimes (\alpha y)$ for all $x \in X$, $y \in Y$, and $\alpha \in \mathbb{F}$. d) $x \otimes 0 = 0$ and $0 \otimes y = 0$ for all $x \in X$ and for all $y \in Y$.

• Proof of a) For all $f \in X^*$ and $g \in Y^*$ we have that:

(1)

\begin{align} \quad [(x_1 + x_2) \otimes y](f, g) = f(x_1 + x_2)g(y) = [f(x_1) + f(x_2)]g(y) = f(x_1)g(y) + f(x_2)g(y) = (x_1 \otimes y)(f, g) + (x_2 \otimes y)(f, g) \quad \blacksquare \end{align}

• Proof of b) For all $f \in X^*$ and $g \in Y^*$ we have that:

(2)

\begin{align} \quad [x \otimes (y_1 + y_2)] = f(x)g(y_1 + y_2) = f(x)[g(y_1) + g(y_2)] = f(x)g(y_1) + f(x)g(y_2) = (x \otimes y_1)(f, g) + (x \otimes y_2)(f, g) \quad \blacksquare \end{align}

• Proof of c) For all $f \in X^*$ and $g \in Y^*$ we have that:

(3)

\begin{align} \quad [(\alpha x) \otimes y)](f, g) = f(\alpha x)g(y) = \alpha f(x)g(y) = \alpha (x \otimes y)(f, g) \end{align}

• And also:

(4)

\begin{align} \quad [\alpha(x \otimes y)](f, g) = \alpha f(x)g(y) = f(x)g(\alpha y) = [x \otimes (\alpha y)](f, g) \quad \blacksquare \end{align}

• Proof of d) For all $f \in X^*$ and $g \in Y^*$ we have that:

(5)

\begin{align} \quad (x \otimes 0)(f, g) = f(x)g(0) = f(x) \cdot 0 = 0 \end{align}

• And also:

(6)

\begin{align} \quad (0 \otimes y)(f, g) = f(0)g(y) = 0 \cdot g(y) = 0 \quad \blacksquare \end{align}