I have a Smart TV that has ethernet but no wireless capabilities.
I also have a spare RPi1 (Model B) with a wireless USB adapter.
Is there anyway I can use the Pi to connect the TV to my wireless router?
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i wished to comment only,
you are asking about bridging the Ethernet adapter to the wifi adapter,
check this, https://www.raspberrypi.org/forums/viewtopic.php?t=132674 it is so user friendly
You can make the RasPi a router that routes data between the wired and wireless interfaces. But I don't believe that the old RasPi with the limitations of USB for ethernet will be strong enough to ensure an uninterruptible video stream. You can try it. How to setup a router you can look at Can a Raspberry Pi Zero W be turned into an USB WiFi dongle to any USB Host like x86 PC or mini-PC? but only use the section: Use routing.