Question

Math 272B Winter 2019 Activity 12 Name: For each of the following subsets of F (0, 1, determine if the the subet is a subspace (and explain). yes no yes no yes no yes no 4, {f I f(z)2 0 for all # e lo, ll} yes no 5. If I f(z +) ()f() for all z, y (0, 1 yes no 6. 1I J is integrable and J fa) dz0

Answer 1

Subspace rightarrow Let V be a vector space over the field if . Now let S is a non - empty subset of v then S is said to be vector subspace of V if O elementof S & x, y elementof S, a, b elementof I F a x + b y elementof S Here V = F [0, 1] . over I R is 2 vector space . S = { f: f (0) = 0 } since O (0) = 0 Rightarrow 0 function belongs in S . Now let f & g elementof S Rightarrow f (0) = g (0) = 0 . Let alpha, beta be arbitrary element of I R . then (alpha f + beta g) (0) = (alpha f) (0) + (beta g) (0) = alpha f (0) + beta g (0) = alpha times 0 + beta times 0 = 0 Rightarrow alpha f + beta g elementof S . Hence S is a subspace of V \r\n as, S = { f: f (0) = 1 } Rightarrow O NotElement