How do I use a single wireless adapter for both an access point and client on Raspberry Pi?

Solution 1:

Using a single WLAN device both as access points (AP) and client (station, STA) is only possible if the device supports it. You can use iw phy or iw list to find out if your device does; there's a line valid interface combinations which describes what combinations are possible (including other mode). Details are for example here.

If your device supports it, you can add new virtual interfaces with something like

iw phy phy0 interface add wlan0_ap type ap

(modify as needed, the other type is sta for client/station).

Then you can run hostapd on one interface, and use the other interface normally.