this tag is for questions about weak Cauchy sequences in the sense of weak topology on a normed linear space.
A sequence of points $ ( x _ n) _ { n= 1 } ^ \infty $ in a normed linear space $X$ is weakly Cauchy whenever for every $x'\in X'$, i.e. every bounded functional $x'$ in the dual space $X'$, the sequence $\big( x'(x_n) \big)_{n=1}^\infty$ is a Cauchy sequence of scalars.
A sequence of points $ ( x _ n )_ { n = 1 } ^ \infty $ in $X$ is weakly convergent whenever there is $x\in X$ such that for every $x'\in X'$, the sequence $\big( x'(x_n) \big)_{n=1}^\infty$ converges to $x'(x)$.
Weakly convergent sequences are weakly Cauchy, and weakly Cauchy sequences are bounded. A subset $E$ of a space $X$ with the property that every sequence of points of $E$ has a weakly Cauchy subsequence, is bounded.
A normed linear space is weakly complete whenever every weakly Cauchy sequence of its points is weakly convergent. Among the most important weakly complete spaces are reflexive spaces.
