The Supreme Court seems to have settled down well at its new home in Parliament Square. In the main courtroom, frosted glass has been installed on the doors behind the judicial bench so that spectators can no longer gaze into one of the judges’ rooms. The judges’ microphones are now kept on all the time so that we can hear every interjection. The court now tweets. Lord Phillips, the retiring president, is even succeeding in his ambition to persuade advocates to follow the same dress code as the justices - who wear robes only on ceremonial occasions. It’s now up to counsel in each case to decide whether they want to turn up in wigs and gowns, gowns alone or normal business clothes; the only requirement is consistency. So Dinah Rose QC decided last week that she had no wish to wear court dress while representing Julian Assange; and Clare Montgomery QC, for the Swedish public prosecutor who is requesting the WikiLeaks founder’s extradition, was equally happy to, er, follow suit.

And what a privilege it was to watch two of the country’s leading advocates slug it out over two days. They referred politely to each other as ‘Miss Rose’ and ‘Miss Montgomery’: flattering euphemisms such as ‘my learned friend’ had, for the most part, gone the way of 18th-century legal costume. Rose certainly had the more principled argument. She maintained that a public prosecutor could not be a judicial authority within the meaning of the Extradition Act 2003. A prosecutor lacked the impartiality and independence that constitute the essential features of judicial authority, Rose argued. The rule that nobody could be a judge in their own cause was fundamental to natural justice. It followed that the European Arrest Warrant (EAW) issued by Marianne Ny, the Swedish prosecutor, was invalid and Assange must be released.

Montgomery had the more practical argument. It was that the term ‘judicial authority’ had a wide and autonomous meaning, covering prosecutors and others who authorised arrests or made extradition requests. That was what the term meant in the EU’s framework decision that preceded - and was brought into force by - the 2003 act. It was also what the term meant in continental Europe. If a judicial authority had to be a judge, a number of our EU partners would have to change their legal systems before any more suspects or fugitives could be extradited from the UK. This was not what EU states had intended when they agreed in 1999 on mutual recognition of judicial decisions.

Rose had to admit that authority was against her, that English courts had always proceeded on the basis that they had no jurisdiction to enquire into the status of the authority that issued an arrest warrant. But her argument tapped into concerns that some European states are seeking - and obtaining - the extradition of British citizens from the UK without the judicial safeguards that are required in Britain. Montgomery accepted that greater scrutiny might be justified if a public prosecutor had issued an EAW without the oversight of a domestic court. But, she maintained, this should not stop courts in the UK from considering such requests.

All these arguments could be followed by anyone with access to a computer in ways that were unthinkable as little as a year ago. Unobtrusive television cameras in each corner of the room allow the entire proceedings to be shown by broadcasters; Sky News now provides a live feed on its website, no doubt in the hope that this will lead, eventually, to the broadcast of criminal trials. The television feed was monitored by some newspapers to provide running reports of the hearing on their websites. These were supplemented by reporters filing reports from within the courtroom. Other journalists were tweeting: there was little risk they would find themselves in contempt of court. Whenever counsel paused for breath, the loudest sound you could hear was the typing of a dozen reporters.

Some of the judges were using computers too, though others preferred the printed bundles. Lord Mance had the best of both worlds, resting heavy papers on a replica oak bookrest from the Bodleian Library shop while touch-typing on a laptop. All this helped make up, to some extent, for the fact that the parties’ written arguments were not available online.

Ironically, the one person who did not have a computer was the WikiLeaks man himself. Sitting next to his solicitor Gareth Peirce, Assange made occasional notes on a pad of yellow stickies. He seemed almost irrelevant to the proceedings, which had nothing to do with whether he deserved to face trial in Sweden on charges of sexual assault and still less to do with the secrets he had once disclosed.

And who won? Seasoned observers say that you can never be sure, especially with a seven-judge court: the real argument begins after the hearing ends. Rose certainly had a more sympathetic audience, though that may not mean much: judges sometimes put the toughest question to the counsel whose argument they favour, if only to persuade waverers on the bench how effective that argument really is.

Equally misleading may be the Supreme Court’s not-so-hidden clue. The judgment appealed against was headed Assange v Swedish Prosecution Authority. That was also the heading that appeared on the written submissions made by each side. But what title did the court give to the respondent on its website and the comprehensive note it supplied to the media? It was ‘Swedish Judicial Authority’. Isn’t that what the court was meant to be deciding?