The Metropolitan Police have defended a decision to downgrade an inquiry into one of the men who carried out Saturday night's terror attack.
They said Pakistan-born Khuram Butt, 27, of Barking, London, was known to police and MI5 in 2015, but there had been no evidence of a plot.
The two other perpetrators were not known to security services.
A minute's silence was held at 11:00 BST on Tuesday to remember victims and those who were affected.
Seven people were killed and 48 injured in the attack which began at 21:58 BST on Saturday night. NHS England said 36 people remained in hospital, with 18 in a critical condition.
All 12 people arrested on Sunday after the London attack have now been released without charge.
On Tuesday, counter-terror officers were searching a property in Ilford, east London, after entering the address at around 01:30 BST. No arrests had been made, police said.
Butt and his two accomplices drove a hired van into pedestrians on London Bridge before stabbing people in the area around Borough Market.
All three men were shot dead by police within eight minutes of receiving a 999 call.
Butt had featured in a Channel 4 documentary The Jihadis Next Door, broadcast last year.
Another of the attackers has been named by police as Rachid Radouane, 30, from Barking. He was a chef who also used the name Rachid Elkhdar and police said he claimed to be Moroccan-Libyan.
What did police know about Khuram Butt?
Butt featured in a Channel 4 documentary last year about Islamist extremists with links to the jailed preacher Anjem Choudary.
The married father-of-two, who worked for London Underground as a trainee customer services assistant for nearly six months last year, could be seen in the programme arguing with police officers in the street, after displaying a flag used by so-called Islamic State in a London park.
Two people in Barking, east London, had also raised concerns about Butt, the BBC's home affairs correspondent Dominic Casciani said.
One man called the anti-terrorism hotline in 2015, and a woman went to the local police because she was scared Butt was trying to radicalise her children.
Metropolitan Police assistant commissioner Mark Rowley said an investigation into Butt began in 2015, but "there was no intelligence to suggest that this attack was being planned and the investigation had been prioritised accordingly".
The inquiry was "prioritised in the lower echelons of our investigative work", Mr Rowley added.
Asked if that had been a poor decision, Mr Rowley said he had seen nothing yet to suggest it, according to the BBC's home affairs correspondent, Danny Shaw.
At any one time there are around 500 active counter-terrorism investigations concerning 3,000 people of interest.
Mr Rowley said work was continuing to understand more about the attackers, "their connections and whether they were assisted or supported by anyone else".
Analysis: Khuram Butt showed his extremist colours
By Dominic Casciani, BBC Home Affairs correspondent
It's still not clear when Khuram Butt got involved in radical Islamist politics, but there is ample evidence that he was involved in the al-Muhajiroun network - certainly in 2015 and potentially at least two years earlier still.
The main evidence comes from his appearance in a Channel 4 documentary, The Jihadis Next Door, broadcast last year.
The film was a close encounter with part of the ALM network and one of its subjects was Siddhartha Dhar, one of Anjem Choudary's right-hand men.
Dhar later skipped bail for Syria. Once there, he appeared in a black mask in an IS execution video.
Butt's links may go back further still. Mohammed Shafiq of the Manchester-based Ramadhan Foundation, an anti-extremism group, says that he believes he was verbally assaulted by Butt in 2013 - the day after another ALM follower killed Fusilier Lee Rigby in Woolwich, south-east London.
Who were the victims?
Canadian national Chrissy Archibald, 30, was the first victim to be named. Her family said she had died in her fiancé's arms after being struck by the attackers' speeding van.
The sister of 32-year-old James McMullan, from Hackney, east London, said he was believed to be among those who died, after his bank card was found on a body at the scene.
A French national was also killed in the attack, according to foreign minister Jean-Yves Le Drian.
The Met have set up a casualty bureau on 0800 096 1233 and 020 7158 0197 for people concerned about friends or relatives.
'You will not win'
A vigil was held at Potters Field Park by the River Thames on Monday evening to remember the victims.
Mayor of London Sadiq Khan led the short ceremony. Addressing the attackers, he said: "We will defeat you. You will not win."
A book of condolences will open on Tuesday at Southwark Council's headquarters in Tooley Street.
Prime Minister Theresa May, who has already signed it, said in her message that British values are "superior to anything offered by the preachers and supporters of hate".
In the days since the attack, Labour has been targeting cuts to the Home Office's policing budget, accusing Theresa May of "letting austerity damage her ability to keep us safe".
Speaking on Tuesday, Mr Khan warned it could be harder to "foil future terrorist attacks" if the Conservatives cut police budgets in London.
"The Conservative plans mean another £400m of cuts to the Met," he said. "I'm simply not willing to stand by and let this happen."
Foreign Secretary Boris Johnson insisted police numbers "remained high" and that the security services did an "incredible job at keeping people safe".
Mr Johnson told BBC Breakfast: "All that argument detracts from the responsibility of those scumbags and what they have done."