Death of a Salesman (Royal Shakespeare Theatre, RSC)

Brooklyn-based Willy Loman is "the New England man" as a travelling sales rep, but he's sixty-three years old and running out of road. His two sons, Biff and Happy, are in their thirties and still at home. His wife Linda is silently coping with his depressions and infidelities. And now he's been taken off salary.

Arthur Miller's tremendous play, dating from 1949 and presenting Willy's crisis as a sour analysis of the post-war American dream, is one any great company, and any great actor, would want to address. Whether or not the RSC, and Antony Sher, should be doing so for a short run on its main Stratford stage – and on Shakespeare's birthday, too – is a matter for another discussion, though the local rumblings of discontent, led by Professor Stanley Wells, are growing.

Director Gregory Doran's defence is that the play has Shakespearean properties of poetry, character and theatrical density, and he makes his case triumphantly in a staging of dissolving locations, cross-fades, musical atmospheres and brilliant, deft touches

At one point the city that has swallowed Willy comes alive as criss-crossing New Yorkers rise on a lift from below the stage – and Sher plays the flawed and deluded anti-hero with a banked-down, candescent energy and the gleaming, but faded, optimism of a little man who's everyman.

Stephen Brimson Lewis's design makes full use of the thrust arena but also reasserts the old proscenium as the Loman household is solidly presented against a backdrop of tenement staircases and windows. Elia Kazan's original legendary production is written into Miller's stage directions and Doran, Brimson Lewis, Tim Mitchell's lighting and Paul Englishby's superb, jazzy and lyrical music come as close to those details, without cravenly reproducing them, as in any production I've seen.

Although the original Broadway Willy Loman was the large and lumbering Lee J Cobb, he's really a small, dapper, quick-spirited operator swamped by accumulated failures and utter exhaustion. Sher stands comparison in this respect with the two other little Willies I've most admired, Dustin Hoffman and Warren Mitchell, both directed by Michael Rudman.

When he smells the coffee, or the trees, and plans his seed garden, or the guest houses he'll build for the boys, Sher's Willy is transfigured, which makes his condition all the more poignant and pathetic. The mortgage is paid, he and Linda – played with a beatific gravitas by Harriet Walter – are ready to move on.

But the great second act humiliations in the boss's office, the restaurant where his sons are trying to pick up girls and finally the Boston hotel room where Biff learns a sort of truth about his father, bulldoze the hope with harsh realities.

Alex Hassell and Sam Marks are well contrasted as the sporty siblings Biff and Happy (though the former shouts too much) and there are fine supporting contributions from Guy Paul as Willy's well-travelled, successful brother, Joshua Richards as the stolid and sensible neighbour Charlie (describing the salesman's armoury of a smile and a shoeshine) and Sarah Parks as the woman in Boston grabbing at snatches of Willy's charm and generosity with his supply of silk stockings.

Death of a Salesman runs at the Royal Shakespeare Theatre in Stratford-upon-Avon until 2 May 2015