Current Attempt in Progress
Determine whether the set equipped with the given operations is a vector space.
For those that are not vector spaces identify the vector space axioms that fail.
The set of all real-valued functions \(f\) defined everywhere on the real line and such that \(f(12) = 0\), with the operations
\((f + g)(x) = f(x) + g(x)\)
\((kf)(x) = kf(x)\)
V is not a vector space, and Axioms 6, 7, 8 fail to hold.
V is not a vector space, and Axioms 4 and 5 fail to hold.
V is not a vector space, and Axioms 1, 2, 3 fail to hold.
V is a vector space.
V is not a vector space, and Axioms 1 and 6 fail to hold.