Question

Suppose ? ⊂ R^? , ? ⊂ R^? are nonempty and open and ? : ? → R^? and ? : ? → R^? . Let ℎ : ? × ? → R ?+? be defined by ℎ(u, v) = (?(u), ?(v)). If ? is continuous at x ∈ ? and ? is continuous at y ∈ ? , then show that ℎ is continuous at (x, y) ∈ ? × ? .

Hint: Note that for any vectors z ∈ R ? and w ∈ R ? , ‖(z, w)‖ = ‖(z, 0) + (0, w)‖ ≤ ‖(z, 0)‖ + ‖(0, w)‖ = ‖z‖ + ‖w‖. And, for any vectors u ∈ R ? and v ∈ R ? , ‖(u, v)‖ ≥ max (‖u‖, ‖v‖).

Answer #1

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Show that GL(2, R) acts transitively on R^2 − {0}. Hint: Let v,
w ∈ R^2 be any
nonzero vectors and consider polar coordinates.

3. a. Consider R^2 with the Euclidean inner product (i.e. dot
product). Let
v = (x1, x2) ? R^2. Show that (x2, ?x1) is orthogonal to v.
b. Find all vectors (x, y, z) ? R^3 that are orthogonal (with
the Euclidean
inner product, i.e. dot product) to both (1, 3, ?2) and (2, 7,
5).
C.Let V be an inner product space. Suppose u is orthogonal to
both v
and w. Prove that for any scalars c and d,...

Suppose K is a nonempty compact subset of a metric space X and
x∈X.
Show, there is a nearest point p∈K to x; that is, there
is a point p∈K such that, for all other q∈K,
d(p,x)≤d(q,x).
[Suggestion: As a start, let S={d(x,y):y∈K} and show there is a
sequence (qn) from K such that the numerical sequence (d(x,qn))
converges to inf(S).] Let X=R^2 and T={(x,y):x^2+y^2=1}.
Show, there is a point z∈X and distinct points a,b∈T
that are nearest points to...

Suppose is orthogonal to vectors and . Show that is orthogonal
to every in the span {u,v}.
[Hint: An Arbitrary w in Span
{u,v} has the
form
w=c1u+c2v.
Show that y is orthogonal to such a
vector w.]
What theorem can I use for this?

Suppose A is a subset of R (real numbers) sucks that both infA
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Prive that:
A. inf(-A) and sup(-A) exist
B. inf(-A)= -supA and sup(-A)= -infA
NOTE:
supA=u defined by: (u is least upper bound of A) for all x in A,
x <= u, AND if u' is an upper bound of A, then u <= u'
infA=v defined by: (v is greatest lower bound of A) for all y in...

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Suppose S ⊂ R is nonempty and M is an upper bound for S. Show M
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For a nonempty subset S of a vector space V , define span(S) as
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containing S. Hint: let W be the intersection of all subspaces
containing S and show W = span(S).
(c) What is the smallest subspace...

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