Test 1

Question 1

Definition of a Bounded Operator

Answer 1

An operator is bounded if there is a C>0 such that ||Tx||<=C||x|| for every x.

Question 2

Equivalent conditions for a bounded operator

Answer 2

An operator is continuous at 0 iff it is continuous for a fixed x iff it is continuous at every x iff it is bounded

Question 3

Norm of an operator T

Answer 3

The norm is the infimum over all C such that ||Tx||<=C||x||, and is also the supremum over ||Tx|| for unit length x.

Question 4

The Hahn-Banach Theorem

Answer 4

Let X be a normed space over R, Y a closed subspace, and f a bounded linear functional on Y. Then there exists a linear functional g from X to R which extends f and maintains its norm.

Question 5

Nowhere Dense

Answer 5

A subset of a metric space is nowhere dense if its closure has an empty interior, equivalently, if the complement of the closure is dense.

Question 6

First Category

Answer 6

A set A of a metric space X is of first category if it can be written as the countable union of nowhere dense sets.

Question 7

Baire Category Theorem V1

Answer 7

Let X be a complete metric space and let U_n be a sequence of dense, open sets. Then the intersection over U_n is dense.

Question 8

Baire Category Theorem V2

Answer 8

Every complete metric space is of 2nd category.

Question 9

Uniform Boundedness

Answer 9

Let X,Y be normed spaces, U a subset of X of 2nd category, and F a collection of functions in B(X,Y) with sup{||Tu||:T in F}

Question 10

Open Mapping Theorem

Answer 10

Let X,Y be Banach Spaces with T:X->Y a bounded surjection. Then T is open.

Question 11

Closed Graph Theorem

Answer 11

Let X,Y be Banach spaces, and T in L(X,Y). Then T is bounded iff the graph of T is closed.

Question 12

Topological Convergence

Answer 12

A net x_a converges to x if for every open set U there is an a0 such that x_a is in U for every a>=a0.

Question 13

Weak Topology

Answer 13

Let X be a normed space. The weak topology is defined by the convergence: x_a converges weakly to x if f(x_a) converges to f(x) for all functionals f.

Question 14

Weak* Topology

Answer 14

The Weak* Topology on the dual of a normed space is defined by the convergence: f_a converges to f if f_a(x) converges to f(x) for every x in X.

Question 15

Banach-Alaoglu Theorem

Answer 15

The closed unit ball in X* is compact.

Question 16

Prove the equivalences for a linear operator T

Answer 16

If T is continuous at 0, then T is continuous at some x0. Through the linearity of T and letting x_n->0 we get that T will be continuous at 0. The continuity of T implies both continuity at an arbitrary x0 and 0. If T is bounded and x_n->0 then Tx_n->0, so T is continuous at 0. Conversely, if T is continuous at 0 we can bound T for unit length x, giving that T is bounded.

Question 17

Show the space B(X,Y) is a normed space with the operator norm, and if Y is a Banach space then so is B(X,Y)

Answer 17

We can quickly verify the 3 properties of a norm by definition of the operator norm, so B(X,Y) is a normed space. Past this, if we take a cauchy sequence in B(X,Y) we may define the limit candidate with limits in Y as Y is complete. Cleverly choosing n, m we can show convergence in B(X,Y).

Question 18

Prove the Factor Theorem

Answer 18

Draw the diagram out, the proof is immediate from the way the maps are defined.