Question

Determine whether the set of all continuous functions on [0,1] satisfying f(0) = 1 constitutes a...

Determine whether the set of all continuous functions on [0,1] satisfying f(0) = 1 constitutes a real linear space under the usual operations associated with elements of the set.

Homework Answers

Answer #1

here I am attaching the solution below

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let C [0,1] be the set of all continuous functions from [0,1] to R. For any...
Let C [0,1] be the set of all continuous functions from [0,1] to R. For any f,g ∈ C[0,1] define dsup(f,g) = maxxE[0,1] |f(x)−g(x)| and d1(f,g) = ∫10 |f(x)−g(x)| dx. a) Prove that for any n≥1, one can find n points in C[0,1] such that, in dsup metric, the distance between any two points is equal to 1. b) Can one find 100 points in C[0,1] such that, in d1 metric, the distance between any two points is equal to...
Determine whether the given set ?S is a subspace of the vector space ?V. A. ?=?2V=P2,...
Determine whether the given set ?S is a subspace of the vector space ?V. A. ?=?2V=P2, and ?S is the subset of ?2P2 consisting of all polynomials of the form ?(?)=?2+?.p(x)=x2+c. B. ?=?5(?)V=C5(I), and ?S is the subset of ?V consisting of those functions satisfying the differential equation ?(5)=0.y(5)=0. C. ?V is the vector space of all real-valued functions defined on the interval [?,?][a,b], and ?S is the subset of ?V consisting of those functions satisfying ?(?)=?(?).f(a)=f(b). D. ?=?3(?)V=C3(I), and...
If R is the ring of all real valued continuous functions defined on the closed interval...
If R is the ring of all real valued continuous functions defined on the closed interval [0,1] and if M = { f(x) belongs to R : f(1/3) = 0}. Show that M is a maximal ideal of R
prove that these functions are uniformly continuous on (0,1): 1. f(x)=sinx/x 2. f(x)=x^2logx
prove that these functions are uniformly continuous on (0,1): 1. f(x)=sinx/x 2. f(x)=x^2logx
Show there does not exist a sequence of continuous functions fn : [0,1] → R converging...
Show there does not exist a sequence of continuous functions fn : [0,1] → R converging pointwise to the function f : [0,1] → R given by f(x) = 0 for x rational, f(x) = 1 for x irrational.
1. Let W be the set of all [x y z}^t in R^3 such that xyz...
1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0. Is W a subspace of R^3? 2. Let C^0 (R) denote the space of all continuous real-valued functions f(x) of x in R. Let W be the set of all continuous functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?
Determine if the following subsets are subspaces: 1. The set of differentiable functions such that f´...
Determine if the following subsets are subspaces: 1. The set of differentiable functions such that f´ (0) = 0 2. The set of matrices of size nxn with determinant 0.
Are the following sets groups: (a) continuous real functions on [0, 1] with operation addition; (b)...
Are the following sets groups: (a) continuous real functions on [0, 1] with operation addition; (b) continuous real functions on [0, 1] with operation multiplication.
3. For each of the piecewise-defined functions f, (i) determine whether f is 1-1; (ii) determine...
3. For each of the piecewise-defined functions f, (i) determine whether f is 1-1; (ii) determine whether f is onto. Prove your answers. (a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0. (b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.
Q 1 Determine whether the following are real vector spaces. a) The set C with the...
Q 1 Determine whether the following are real vector spaces. a) The set C with the usual addition of complex numbers and multiplication by R ⊂ C. b) The set R2 with the two operations + and · defined by (x1, y1) + (x2, y2) = (x1 + x2 + 1, y1 + y2 + 1), r · (x1, y1) = (rx1, ry1)