# (X⊗pY)* and BL(X, Y; F) are Isometrically Isomorphic

Recall from The Projective Tensor Product of X⊗Y page that if $X$ and $Y$ are normed linear spaces then we defined the projective tensor norm on $X \otimes Y$ for all $u \in X \otimes Y$ by:

(1)Where the infimum above varies through all finite representations of $U$ as elements in $X \otimes Y$. We prove that the projective tensor norm is indeed a norm on $X \otimes Y$ and we defined the projective tensor product of $X$ and $Y$ to be the completion $(X \otimes_p Y, p)$. We also proved that if $w$ denotes the weak projective norm on $X \otimes Y$ then $w(u) \leq p(u)$ for all $u \in X \otimes Y$ and furthermore, $p(x \otimes y) = \| x \| \| y \|$.

We will now prove an important result which states that the dual space $(X \otimes_p Y)^*$ is isometrically isomorphic to $\mathrm{BL}(X, Y; \mathbf{F})$.

Theorem 1: Let $X$ and $Y$ be normed spaces. For each $F \in (X \otimes_p Y )^*$ let $\phi_F : X \times Y \to \mathbf{F}$ be defined for all $x \in X$ and all $y \in Y$ by $\phi_F(x, y) = F(x \otimes y)$. Let $T : (X \otimes_p Y)^* \to \mathrm{BL}(X, Y; \mathrm{F})$ be defined for all $F \in (X \otimes_p Y)^*$ by $T(F) = \phi_F$. Then $T$ is an isometric isomorphism from $(X \otimes_p Y)^*$ to $\mathrm{BL}(X, Y; \mathbf{F})$. |

**Proof:**There are a few things to verify.

**1. Showing that for all $F \in (X \otimes_p Y)^*$ we have that $\phi_F \in \mathrm{BL}(X, Y; \mathbf{F})$:**Let $F \in (X \otimes_p Y)^*$ and consider $\phi_F$. We want to show that $\phi_F$ is a bounded bilinear map on $X$ and $Y$ to $\mathbf{F}$.

- First, fix $x \in X$ and let $y_1, y_2 \in Y$, $\alpha \in \mathbf{F}$. Then by the linearity of $F$ we have that:

- So for each fixed $x \in X$ the map $y \to \phi_F(x, y)$ is linear. It can similarly be shown that for each fixed $y \in Y$ the map $x \to \phi_F(x, y)$ is linear. Furthermore, the outputs of $\phi_F$ are elements of $\mathbf{F}$. So for all $F \in (X \otimes_p Y)^*$ we have that $\phi_F$ is linear. Furthermore we have that for all $x \in X$, $y \in Y$, since $F$ is bounded:

- So $\phi_F$ is bounded. Thus for all $F \in (X \otimes_p Y)^*$ we have that $\phi_F \in \mathrm{BL}(X, Y; \mathbf{F})$.

**2. Showing that $T : (X \otimes_p Y)^* \to \mathrm{BL}(X, Y; \mathbf{F})$ is linear:**Let $F, G \in (X \otimes_p Y)^*$ and let $\alpha \in \mathbf{F}$. We first want to show that $T(F + G) = T(F) + T(G)$, that is, $\phi_{F + G} = \phi_F + \phi_G$. We then want to show that $T(\alpha F) = \alpha T(F)$, that is, $\phi_{\alpha F} = \alpha \phi_F$.

- So let $x \in X$, $y \in Y$. Then:

- So indeed, $\phi_{F + G} = \phi_F + \phi_G$. Now let $x \in X$ and $\alpha \in \mathbf{F}$. Then:

- So indeed $\phi_{\alpha F} = \alpha \phi_F$. Therefore $T$ is linear.

**3. Showing that $T$ is bounded with $\| T \| \leq 1$:**For all $F \in (X \otimes_p Y)^*$ we have that:

- So $T$ is bounded with $\| T \| \leq 1$.

**4. Showing that $\| F \| \leq \| T(F) \|$ for all $F \in (X \otimes_p Y)^*$:**Let $\phi \in \mathrm{BL}(X, Y; \mathbf{F})$. By the theorem on The Existence of a Linear Map σ on X⊗Y to Z that Matches a Bilinear Map on X×Y to Z page there exists a bilinear map $F : X \otimes Y \to \mathbf{F}$ for which $F(x \otimes y) = \phi(x, y)$ for all $x \in X$, $y \in Y$. Note that:

- So $|F(u)| \leq \| \phi \| \sum_{i=1}^{m} \| x_i \| \| y_i \|$ for all representations $\sum_{i=1}^{m} x_i \otimes y_i$ of $u$, and so, $|F(u)| \leq \| \phi \| p(u)$. Hence for all $u \in X \otimes Y$ we have that $|F(u)| \leq \| \phi \| p(u)$, so:

- Since this holds for all $\phi \in \mathrm{BL}(X, Y; \mathbf{F})$, it certainly holds for all $T(F) = \phi_F \in \mathrm{BL}(X, Y; \mathbf{F})$, and so $\| F \| \leq \| T(F) \|$ for all $F \in (X \otimes_p Y)^*$.

- Thus we conclude that $T : (X \otimes_p Y)^* \to \mathrm{BL}(X, Y; \mathbf{F})$ is an isometric isomorphism of $(X \otimes_p Y)^*$ to $\mathrm{BL}(X, Y; \mathbf{F})$. $\blacksquare$

We state a more general version of the above theorem below:

Theorem 2: Let $X$, $Y$, and $Z$ be normed linear spaces. Then $\mathcal B (X \otimes_p Y, Z)$ is isometrically isomorphic to $\mathrm{BL}(X, Y; Z)$. |

*Here, $\mathcal B(X \otimes_p Y, Z)$ denotes the space of all bounded linear operators from the projective tensor product $X \otimes_p Y$ to $Z$, and $\mathrm{BL}(X, Y; Z)$ denotes the space of all bounded bilinear mappings from $X \times Y$ to $Z$. When $Z = \mathbf{F}$ we obtain Theorem 1.*