MATHEMATICS 421/510, PROBLEM SET 1
Due on Thursday, January 19
Write clearly and legibly, in complete sentences. You may discuss the
homework with other students, but the final write-up must be your own. If
your solution uses any results not introduced in class, state the result clearly
and provide either a reference or a proof.
1. (10 points for each part) Let X be a linear space over F and let Y be
a subspace of X. We define X/Y to be the set of equivalence classes
[x] = {x0 ∈ X : x0 − x ∈ Y }, x ∈ X.
(a) Prove that the operations [x]+[x0 ] = [x+x0 ] and c[x] = [cx] are well
defined, and the X/Y is a linear space over F with these operations.
(b) Let T : X → Z be a linear mapping, where Z is another linear
space over F. Prove that the mapping T 0 : X/NT → RT defined by
T 0 [x] = T x is well defined, linear, one-to-one and onto.
(c) Let ` : X → F be a linear functional such that ` 6≡ 0. Prove that
the quotient space X/N` has dimension 1.
2. (10 points) Let X be a linear space over R. Recall that x is an interior
point of a set S ⊂ X if for every y ∈ X there is an > 0 such that
x + ty ∈ S for all |t| < .
Let K be a convex set such that 0 is its interior point, and let pK (x) =
inf{a > 0 : x ∈ aK}. Prove that x is an interior point of K if and
only if pK (x) < 1.
3. (10 points for each part) An extreme point of a convex set K ⊂ X,
where X is a linear space, is a point x ∈ K such that if x = (y + z)/2
with y, z ∈ K, then y = z = x.
Let X = C([0, 1]) be the linear space of
continuous
R 1 all real-valued
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functions on [0, 1], and let K = {f ∈ X : 0 (1 + f (x)) dx ≤ 1}.
(a) Prove that K is convex.
(b) Does the set K have any extreme points? If so, find them.
R1
(c) Prove that `x : f → f (x) and `∗ : f → 0 f (x)dx are linear
functionals on X.
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(d) Find a linear functional ` on X such that `(f ) < `(g) for all f ∈ K,
where g(x) ≡ 1 on [0, 1].
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