Question

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.

Answer #1

here I am attaching the solution below

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, 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 [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

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 = 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´ (0) = 0
2. The set of matrices of size nxn with determinant 0.

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.

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.

Let S be the set R∖{0,1}. Deﬁne functions from S to S by
ϵ(x)=x, f(x)=1/(1−x), g(x)=(x−1)/x.
Show that the collection {ϵ,f,g} generates a group under
composition and compute the group operation table.

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