What is the definition of a weakly compact operator from a Banach space to another?

Question

What is the definition of a weakly compact operator from a Banach space $X$ to another $Y$? I think the definition needs to know the definition of weak topology but I don't know what is the the weak topology.

Answer 1

Weakly compact operators were introduced by Kakutani $[1]$ and Yosida $[2]$ when they proved a Banach space version of the mean ergodic theorem: if the norms $||T^n||$ of $T$ linear and weakly compact operator are uniformly bounded, then the averages $$\frac{x + Tx + \cdots T^{n-1}x}{n}$$ converge in norm to a limit $Px$ for all $x \in X$, and $x \mapsto Px$ defines a projection from $X$ onto $T$'s set of fixed points.

A linear operator $T : X \rightarrow Y$ is called weakly compact if it takes bounded (in norm) subsets of $X$ to weakly relatively compact subsets of $Y$. From the algebraic point of view, weakly compact operators form a closed ideal.

Assume now that $X$ and $Y$ are Banach spaces. By the so-called Eberlein–Šmulian theorem, $T : X \rightarrow Y$ is weakly compact if and only if every bounded sequence in $X$ admits a subsequence whose image is weakly convergent in $Y$.

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$[1]$ S. Kakutani, Iteration of linear operations in complex Banach spaces, Proc. Imp. Acad. Tokyo 14 (1938), 295-300

$[2]$ K. Yosida, Mean ergodic theorem in Banach spaces, Proc. Imp. Acad. Tokyo 14 (1938), 292-294.