Recent content by cummings12332

Homework Statement Let F be the field and f(x)=x-1,g(x)=x^2-1 and F[x]/(f(x)) is isomorphism to F, is it g(x) maximal?? 2. The attempt at a solution I will say no.Since g(x) is not 0, the dieal (x^2-1) in a prime idea domain F is maximal iff (x^2-1) is irreducible. And we say...

Homework Statement Is (x^2-1) a unit in F[x]? where F is a field. 2. The attempt at a solution I might say yes, cause we can find the taylor expansion of 1/(x^2-1), is my idea right?

Can I choose the radius of open ball be r=min{1-x1,x1} then for all x=(x1,x2) in A we have B(x,r) is contained in A so A is open??

If I choose the centre be (1,1) and radius 1 to be the open ball then it is segment with length 2 , 45degree to x-axis and the midpoint of the segment is (1,1)

Homework Statement can someone help me to solve these problems in details?? Consider A =(0,1)× R. Is A open w.r.t. the topology induced by the French railway metric in R2? how about B=(-1,1)× R? 2. The attempt at a solution I know A is open in the topology induced by d if and only...

omg! that's what i thought at the beginning! many thanks !

for me ,it is obviouse. 0\leq f_n\leq 1. then max fn =1, min fn =0 and limfn =f so by sanwich rule 0\leq f(x)\leq 1 i just don't know how should i prove that f is bounded by [0,1] here

i not sure how to prove that 0<f<1 ,which way should i begin with??

δ if f_n is a convergent sequence in A such that f_n\rightarrow f, then for f_n is in A , A is a subset of C[0,1] so fn is bounded by 0 and 1 and continuous on [0,1] so fn is uniformly continuous to f ,then f is bouded by 0,1 which is in A? and secound part if i use the definition of...

Homework Statement The Attempt at a Solution for part A i assume that f is in norm space C[0,1],||.|| , then choose a sequence fn in C[o,1] s.t fn->f then for 0<fn<1 so 0<f<1 i.e. A is closed i am not sure my answer here for part B i assume the anti-derivatice of f(t) to be...

how to say that \Theta(f) is continuous? i just don't know how to prove here. if it is then i know how to solve the problem now,many thanks

Homework Statement The Attempt at a Solution set x(t)=1+∫2cos(s(f^2(s)))ds(from 0 to t) then check x(0)=1+∫2cos(s(f^2(s)))ds(from 0 to 0)=1 then the initial condition hold, by FTC, we have dx(t)/dt=2cos(tx^(t)), then solutions can be found as fixed points of the map but for...

Many thanks, I have solved the first part out. but for secound part i still don't know how to begin. should i choose a sequence fn in A then check that ||fn-f||<esillope,then what should i do for it?could u give me more details?

Homework Statement show the set {f: ∫f(t)dt>1(integration from 0 to 1) } is an open set in the metric space ( C[0,1],||.||∞） and if A is the subset of C[0,1] defined by A={f:0<=f<=1} is closed in the norm ||.||∞ norm. Homework Equations C[0,1] is f is continuous from 0 to 1.and ||.||∞...