functional analysis - Can we embed $X'otimes Y$ into the space of bounded, linear operators $Xto Y$?

Let

• $\mathbb F\in\left\{\mathbb C,\mathbb R\right\}$



• $X$ and $Y$ be normed $\mathbb F$-vector spaces


• $X'$ denote the topological dual space of $X$


• $\mathfrak L(X,Y)$ denote the space of bounded, linear operators from $X$ to $Y$


• $\mathfrak B(X'\times Y,\mathbb F)$ be the space of bilinear forms on $X'\times Y$


• $X\otimes Y$ denote the tensor product of $X$ and $Y$

Can we show that $X'\otimes Y$ can be embedded into $\mathfrak L(X,Y)$, i.e. that there is a

1. injective,


2. continuous and


3. open

mapping $\iota:X'\otimes Y\to\iota(X'\otimes Y)$?

Clearly, we would need to choose a norm on $$X'\otimes Y:=\operatorname{span}\left\{\varphi\otimes y:(\varphi,y)\in X'\times Y\right\}\;,$$ where $$(\varphi\otimes y)(A):=A(\varphi,y)\;\;\;\text{for }B\in\mathfrak B(X'\times Y,\mathbb F)\;.$$ I think that the projective norm $$\pi(u):=\inf\left\{\sum_{i=1}^n\left\|\varphi_i\right\|_{X'}\left\|y_i\right\|_Y:u=\sum_{i=1}^n\varphi_i\otimes y_i\right\}$$ will do it.

My idea is to define $$(\iota u)(x):=\sum_{i=1}^n\varphi_i(x)y_i\;\;\;\text{for }x\in X\tag 1$$ for $u\in X'\otimes Y$ with $u=\sum_{i=1}^n\varphi_i\otimes y_i$.

This $\iota$ is obviously linear. Maybe we can show that it is bounded too (i.e. a bounded, linear operator). This would yield (2.). How can we show this and how can we show (1.) and (3.)?