My background in linear algebra is pretty basic: high school math and a first year course about matrix math. Now I'm reading a book about finite-dimensional vector spaces and there are a few concepts that are just absolutely bewildering to me: dual spaces, dual bases, reflexivity and annihilators. The book I'm reading explains everything in extremely general terms and doesn't provide any numerical examples, so I can't wrap my head around any of this. I'd really appreciate it if my loose understanding of these concepts could be critiqued/corrected and, if possible, some simple numerical examples could be provided. I really can't make heads or tails of some of this. Note: I've never seen this bracket notation before, so I'll briefly introduce it in case it isn't something standard: [x,y][itex]\equiv[/itex]y(x) First, dual spaces. My understanding of a linear functional is that it's a black box where vectors go in and scalars come out (e.g. dot product). The dual space V' of a vector space V, is the set of all linear functionals that can be applied to that vector space. So, why is this called a "space"? How can things like integration and dot products (i.e. operations) form a space? The author also refers to the elements of V' as "vectors" -- how can an operation be a vector? My understanding of a vector is that it is a value with both magnitude and direction. Obviously, operations produce values, but V' is the set of operations, not values. Second, dual bases. I just don't understand this at all, so I'll just provide the definition in this book: I think ∂ij is the Kronecker delta, but I'm not 100% sure. So, what I think this means is that there is one operation in V' for each V, for which yj(xi)=1 for j=i and 0 for all j≠i. But does V' necessarily have dimension n? Third, reflexivity. I just don't understand this at all. Here's the definition in the book: Fourth, annihilators. I think I understand this concept somewhat, but the proofs presented don't make sense to me. My understanding of an annihilator is that it is any subset of V' which evaluates to 0 for all x in V. The thing I'm confused about is the annihilator of an annihilator. Now, I'm willing to accept this proposition, but the proof is relatively short yet I cannot make sense of it. The proof is: The problem I have here is "By definition, M00 is the set of all vectors x such that [x,y]=0 for all y in M0". Shouldn't it be the set of all vectors z (or whatever letter you like) in V'?