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It's a £75 million project being set up and run by a charity, the Garden Bridge Trust, so it won't be publicly owned. It will be closed to the public when it's rented out for commercial purposes.

The FAQ says:

Over 65% of the capital costs to build the bridge will be fundraised from the private sector. More than £145 million has been pledged already and there is a business plan to cover the £2 million annual maintenance and operations costs. Transport for London and the Government have together contributed £60 million in total.

https://www.gardenbridge.london/questions-answers/fact-ficti...

The arguments for the public funding contribution are that commuters will be able to walk across it, and it will help regenerate the areas at both ends.