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The Gowers dichotomy theorem states that every infinite-dimensional Banach space contains either an unconditional basic sequence or a hereditarily indecomposable subspace. Choose an initial basic sequence; it suffices to prove the result for its closed span $X$, with basis constant $K$.

We use the following precise finite-sequence form of the Gowers Ramsey theorem for Banach spaces. Let $\mathcal{A}$ be an analytic family of finite normalized block sequences, large in the sense that every infinite-dimensional block subspace contains a member. Given positive errors $\Delta =({\delta }_i)$, some infinite-dimensional block subspace $Y$ makes ${\mathcal{A}}_{\Delta }$ strategically large. Here ${\mathcal{A}}_{\Delta }$ consists of tuples within ${\delta }_i$ coordinatewise of a member of $\mathcal{A}$. In the block game the subspace player chooses an infinite-dimensional block subspace at each turn; the vector player chooses a successive unit vector in it and can force a finite initial segment into ${\mathcal{A}}_{\Delta }$. This is the Ramsey theorem permitted without proof, not the dichotomy conclusion.

Assume $X$ contains no unconditional basic sequence. For $\epsilon >0$, let ${\mathcal{A}}_{\epsilon }$ be the finite normalized block tuples $(x_1,\dots ,x_n)$ for which the odd and even spans contain unit vectors $u,v$ with $\Vert u-v\Vert <\epsilon$. This family is analytic: in each finite length the witness coefficients and vectors range over finite-dimensional scalar spaces, and the strict distance test is continuous; taking a countable union of finite-support choices gives the block family.

It is large. Indeed, every block subspace has a conditional block basis, with coordinate projections of unbounded norms. Given a finite combination $z$, choose a set of coordinates $A$ with $\Vert P_{A}z\Vert$ arbitrarily large relative to $\Vert z\Vert$. Put $u_0=P_{A}z$, $v_0=-P_{A^c}z$. Then $u_0-v_0=z$ and $|\Vert u_0\Vert -\Vert v_0\Vert |\le \Vert z\Vert$, soGroup consecutive nonzero terms belonging to the same side into a single block, and normalize each group. The groups alternate between the two sides, and their odd and even spans contain these two normalized vectors. Interchanging their names if necessary gives a member of ${\mathcal{A}}_{\epsilon }$.

Choose $\Delta$ with $8K{\sum }_i{\delta }_i<\epsilon$. A game tuple in ${\mathcal{A}}_{\epsilon ,\Delta }$ actually belongs to ${\mathcal{A}}_{2\epsilon }$. To check the perturbation explicitly, each coefficient of a unit vector in a normalized block tuple is bounded by $2K$. Replacing its coordinates by their nearby game vectors changes it by at most $2K\sum {\delta }_i$. Normalizing again changes the vector by at most twice that amount. Doing this for the two witnesses adds at most $8K\sum {\delta }_i$ to their distance.

The Ramsey theorem thus supplies a block subspace on which the vector player can force odd and even spans to contain unit vectors within $2\epsilon$. Given any two infinite-dimensional block subspaces $U,V$ of that space, let the subspace player choose them alternately, using their successive tails to satisfy support order. The strategy produces the odd vectors in $U$ and the even vectors in $V$, so these subspaces contain the required nearby unit vectors.

Repeat inside decreasing block subspaces $Y_n$ with ${\epsilon }_n\downarrow 0$, and choose a diagonal normalized block sequence $(z_n)$ with $z_n\in Y_n$. Its tail from $n$ onward lies in $Y_n$. In $Z=[z_n]$, any two infinite-dimensional block subspaces can be restricted to this tail, hence contain unit vectors at distance less than $2{\epsilon }_n$ for every $n$.

Finally this zero-angle statement extends to arbitrary infinite-dimensional closed subspaces $U,V$ of $Z$. Choose unit vectors in each subspace annihilating the first successively many basis coordinates, and approximate them by finite successive blocks with summably small errors. The coordinate bound $2K$ makes the resulting map from the block span to the original subspace arbitrarily close to the identity in operator norm. Apply the block-subspace zero-angle result and normalize the transported vectors; their added error can be made arbitrarily small. Thus the unit spheres of $U,V$ have distance zero.

By the equivalence proved in part (a), $Z$ is hereditarily indecomposable. This proves the dichotomy, including the passage from block subspaces to all subspaces.