1. The problem statement, all variables and given/known data First, I'd like to say that this question is from an Introductory Linear Algebra course so my knowledge of vector space and subspace is limited. Now onto the question. Q: Which of the following are subspaces of F(-∞,∞)? (a) All functions f in F(-∞,∞) for which f(0) = 0 (b) All functions f in F(-∞,∞) for which f(0) = 1 (c) All functions f in F(-∞,∞) for which f(-x) = f(x) (d) All polynomials of degree 2 2. Relevant equations Definitions: A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication deﬁned on V. Theorems: If W is a set of one or more vectors in a vector space V, then W is a subspace of V if and only if the following conditions are satisﬁed. ( a ) If u and v are vectors in W, then u + v is in W. ( b ) If k is a scalar and u is a vector in W, then ku is in W. Subspaces: Pn < C^∞ (-∞,∞) < C^m (-∞,∞) < C^1 (-∞,∞), < C(-∞,∞) < F(-∞,∞) (as defined by a diagram in my book) 3. The attempt at a solution I am absolutely stuck. I have read through the chapter but it is general solutions with vague wording that tells you to prove it yourself. "We leave it for you to convince yourself that the vector spaces discussed in Examples 7 to 10 are“nested” one inside the other as illustrated in Figure 4.2.5. Copyright | Wiley | Elementary Linear Algebra | Edition 11". I know it's my lack of understanding of subspaces and vector spaces that is preventing me from doing this kind of problem. I understand the R3 space somewhat but I'm stuck on transferring what I know from that vector space to this function vector space. I understand that a subspace has to be defined under the addition and scalar multiplication of that set. But I do not know how that applies to these functions. A previous question I worked out to show my logic in thinking, (it is probably flawed so please point out my errors): Determine Which of the following are subspaces of R3: (a,0,0) this is a subspace of R3 because it has the zero vector for a=0 and can be added to itself under a= 0 to get 0 also multiplying any real number to this vector if a= 0 still gives 0 (a,b,c) where b= a + c + 1 not a subspace of R3 because it fails to have the zero vector for any a or c that influences b test: for a = 0 c = 0 b = (0) + (0) + 1 =1 a=-1 c=0 b= -1 + 0 +1 =0 So this is my thought process for these problems.