JEREMY Corbyn’s keynote speech at the Labour Party conference in Liverpool was a major improvement on his first as Labour leader.

With his mandate reinforced, he has grown more confident, stumbles less, and appears to be enjoying his position more. But that does not diminish the task facing him.

Mr Corbyn says Labour can take its core values and apply them to 21st century problems. A call for education for all rather than the few, and renationalising the railways might well prove popular. A crackdown on tax evasion is a clear crowd pleaser. But there are major questions about whether Mr Corbyn can win voter support for repeal of the Trade Union Act, or persuade business of the need for higher tax, even in return for investment in infrastructure.

There is still little evidence that he can convince the wider electorate. If he is to have the slightest chance of reaching beyond the Labour supporters in the conference hall he will have to convince voters that the division in the party is at an end and form a credible shadow cabinet.

He called for the party to turn its back on the ‘trench warfare’ which has consumed it for a year and turn its fire on the Conservatives. While the conference has shown there is still resistance to Mr Corbyn from within his own ranks, there were indications after the Labour leader spoke that some opponents may now rally behind him, Hilary Benn backed house-building plans and other non-Momentum MPs tweeted praise for the speech.

Mr Corbyn faces a problem over immigration, with many convinced - even within his own circle - Labour must address the concerns of its traditional supporters. Arguing, as he does, that the issue is low pay and job insecurity rather than immigrants is valid. But it may not be easy to persuade voters.

Electorally, Corbyn says he can climb the mountain, but it looks insurmountable still, and is little less precipitous for the party in Scotland. Kezia Dugdale’s success in establishing the autonomy of Scottish Labour is significant and overdue.

The party has been unable to make headway since devolution, against a perception that key decisions, and policy were framed by the party machine in London.

But the ability to choose candidates with a different view - on Trident, for example - than that taken by the national party, could well spell trouble ahead.

Meanwhile there is confusion over Mr Corbyn’s position on a pact with the SNP. Some within Labour appear to favour a “progressive alliance”, but Ms Dugdale has categorically ruled out such a pact.

The Scottish Labour leader herself faces different challenges, but equally problematic. The SNP and the Greens both offer voters a left of centre alternative, while the Conservatives are in the ascendant and have taken Labour’s place as the official Holyrood opposition.

Greater autonomy gives the Scottish party a chance to forge a distinctive vision for Scotland, while a seat on Labour’s NEC allows Ms Dugdale to declare that she is standing up for Scotland’s interests within the union. But the two leaders have to find a way to work together, on both sides of the border, to revive both parties, to subtly different agendas. It is a very tall order.