Which of the following subsets are subspaces of the vector space C(-∞, ∞) and why or why not? V is the set of all real-valued continuous functions defined on R1. If f ang g are in V we define f+g by (f+g)(t) = f(t) + g(t). If f is in V and c is a scalar, we define c . f by (c . f)(t) = cf(t). Then V is a vector space which is denoted by C(-∞, ∞).
a) All nonnegative functions
b) All constant functions
c) All functions f such that f(0) = 0
d) All functions f such that f(0) = 5
e) All differentiable functions
I'm lost on this and does anybody know a good website to learn about subspaces and subsets. My book is not that great. Thanks!
Do you know what it takes to confirm that a subset is a subspace?
For each pair of vectors u & v in the set and each scalar a you must show that au + v is also in the set.
To get you started. If f is a non-negative function what can you say about (-1)f +0?
Does that say that (a) is not a subspace.