I was studying about Compact Bilinear Operators here. I found the following definition:
Definition Let $X,Y,Z$ be normed linear spaces. A Bilinear Operator $T:X\times Y\to Z$ is said to be compact if $T(B_X\times B_Y)$ is precompact in $Z$.
If I consider $X,Y,Z$ to be reflexive Banach spaces, does the above definition reduces to the following:
Query A Bilinear operator $T:X\times Y\to Z$ is compact if and only if for every pair of weakly convergent sequences $x_n\rightharpoonup x_0$ and $y_n\rightharpoonup y_0$, there holds $T(x_n,y_n)\to T(x_0,y_0)$.
Also, I donot understand the meaning of precompact here.
Thanks in advance.