The Lightning Network is said to be able to take the transactions per second figure of Bitcoin and other cryptocurrencies to unprecedented heights of at least 1 million transactions per second.
Note that “1 million” isn't something fixed. It's a decentralized network and you can't know for sure how many transactions it can hold. The fact that it doesn't have a limit makes it even more interesting. I guess that the more LN nodes, the more transactions it can hold per second.
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Can't I send you money through LN without having to open a channel? For example, a family of four wouldn't have to open four different payment channels, the father could open one and then send off-chain funds to the other members.