Notation in definition of Fréchet differentiation

Question

Below I paraphrase what I found on Wikipedia.

Let $V,W$ be normed vector spaces, and $U\subset V$ be an open subset of $V$. A function $f:U \to V$ is called Fréchet differentiable at $x\in U$, if there exists a bounded linear $A$ operator such that: $$ \lim_{\|h\| \to 0} \frac{\|f(x+h)-f(x)-Ah\|_W}{\|h\|_V}=0 $$

If there exists such an operator $A$, it is unique, so we write $Df(x)=A$. Now, consider a function $f$ that is Fréchet differentiable at for any point of $U$. Then, if the function $$Df:U\to B(V,W)\; ;\; x\mapsto Df(x) $$ is continuous, then the function $f$ is said to be $C^1$.

My question is, what is the meaning of $B(V,W)$ ?

Answer 1

$B(V,W)$ consists of all bounded, linear maps from $V$ to $W$. That is, $T\in B(V,W)$ means that $T:V\rightarrow W$ is linear, and $$\|Tx\|_W\leq C\|x\|_V$$ for all $x\in V.$