Prove that $\ell(\Bbb{R}^n,\ell(\Bbb{R}^m,\Bbb{R}^q))=\ell_b(\Bbb{R}^n\times\Bbb{R}^m,\Bbb{R}^q)$

Question

Prove that \begin{align}\ell(\Bbb{R}^n,\ell(\Bbb{R}^m,\Bbb{R}^q))=\ell_b(\Bbb{R}^n\times\Bbb{R}^m,\Bbb{R}^q)\end{align}

where $\ell(\Bbb{R}^n,\ell(\Bbb{R}^m,\Bbb{R}^q))$ represents the space of continuous linear maps from $\Bbb{R}^n$ to $\ell(\Bbb{R}^m,\Bbb{R}^q)$ and $\ell_b(\Bbb{R}^n\times\Bbb{R}^m,\Bbb{R}^q),$ the space of bilinear maps from $\Bbb{R}^n\times\Bbb{R}^m$ to $\Bbb{R}^q.$ Also $\dim \Bbb{R}^n,\dim \Bbb{R}^m,\dim \Bbb{R}^q<\infty.$

How do I go about this? Any help?

Answer 1

If $T$ is in the left hand side then $S(x,y)=(Tx)(y),x\in \mathbb R^{n},y\in \mathbb R^{m}$ defines an element of the right hand side. If $S$ is given, then for each $x\in \mathbb R^{n}$ define $Tx:\mathbb R^{m} \to \mathbb R^{q}$ by the same equation. All linear maps on finite dimensional spaces are continuous, so you don 't have to verify continuity.