I was in directx 11 and was wondering is local space the same as object space and if not, what is object space?
"Object space" is generally used to refer to the coordinate space that a model or collection of geometry is defined within. Another way to think of this would be the space that the object was created in, where the origin is some useful point within or relative to the object, such as its center, or the place where the model would "touch the ground" when positioned in the world, et cetera.
"Local space" is sometimes synonymous with object space, if you're talking geometry. However, in some specialized contexts it could also be used to refer to any other coordinate system if there is an important distinction to be made between "this" system (the local one) and "that" system (the other one). However this will generally be made apparent explicitly or from the context.
Assuming "local space" refers to "object space" is probably a generally safe assumption. "Model space" is another term you might here.