Let X : = Πα∈IXα be a product space (with
the product topology), πα : X → Xα be the
projection map for each α∈I, and {xn}
be a sequence in X. Prove that the sequence {xn}
converges to a point x∈X if and only if
{πα(xn)} converges to πα(x) for
every α∈I.
Let X : = Πα∈IXα be a product space (with the product topology), πα : X...
Let
be a metric space and let
be the topology on
induced by
, and let
be a compact space. Prove that
is compact.
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Let
be the real line with Euclidean topology. Prove that every
connected subset of
is an interval.
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Let V be a Hilbert space. Let S1 and S2 be two hyperplanes in V defined by Let be given. We consider the projection of y onto , i.e., the solution of (1) (a) Prove that is a plane, i.e., if , then for any . (b) Prove that z is a solution of (1) if and only if and (2) (c) Find an explicit solution of (1). ( d) Prove the solution found in part (c) is unique. We...
Let
be an inner product space (over
or
), and
. Prove that
is an eigenvalue of
if and only if
(the conjugate of
) is an eigenvalue of
(the adjoint of
).
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Let
be a sequence of random variables, and let Y be a random
variable on the same sample space. Let An(ϵ) be the
event that |Yn − Y | > ϵ. It can be shown that a
sufficient condition for Yn to converge to Y w.p.1 as
n → ∞ is that for every ϵ > 0,
(a) Let
be independent uniformly distributed random variables on [0, 1],
and let Yn = min(X1, . . . , Xn).
In class,...
A function from N to a space X is a sequence n-xn in X. A sequence in a topological space converges to a point x E X if for each open neighborhood U of x there exists a є N such that Tn E U for all n 2 N. c) Consider the (non-Hausdorff) space S1,2,3 equipped with the indiscrete topology; that is, the only open sets are and S. Let n sn be an arbitrary sequence in S. Show...
Problem 1. Let (X, d) be a metric space and t the metric topology on X. (a) Fix a E X. Prove that the map f :(X, T) + R defined by f(x) = d(a, x) is continuous. (b) If {x'n} and {yn} are Cauchy sequences, prove that {d(In, Yn)} is a Cauchy sequence in R.
Let
be an orthonormal set of a Hilbert space. Let
and
be two vectors in H. Show that
converges absolutely, and that
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Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...
Let X and Y be a first countable spaces. Prove that f:XY
is continuous if whenever xnx
in X then f(xn
)f(x)
in Y
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