## PROBLEM 1

(A) Define precisely the notion of Riemann integrability for a function $f ( x )$ on [0, 1].

(B) Suppose that $f _ { n } ( x )$ is a sequence of Riemann integrable functions on [0, 1] such that $\{ f _ { n } ( x ) \}$ converges uniformly to $f ( x )$ . Prove that $f ( x )$ is Riemann integrable.

## PROBLEM 2

Let $F ( x ) = \sum _ { n = 0 } ^ { \infty } a _ { n } x ^ { n }$ be a power series with $a _ { n } \in \mathbf { R }$ . Show that there exists a unique number $\rho \geq 0$ such that $F ( x )$ converges if $| x | < \rho$ and $F ( x )$ diverges if $| x | > \rho$

## PROBLEM 3

Prove that the series

$$
f ( x ) = \sum _ { n = 1 } ^ { \infty } { \frac { \sin n x } { n ^ { 5 / 2 } } }
$$

converges for all $x \in \mathbf { R }$ and that $f ( x )$ is a continuous function on $\mathbf { R }$ with a continuous derivative.
State clearly any facts you assume.

## PROBLEM 4

The point $P = ( 1 , 1 , 1 )$ lies on the surface S in $\mathbf { R } ^ { 3 }$ defined by

$$
x ^ { 2 } y ^ { 3 } + x ^ { 3 } z + 2 y z ^ { 4 } = 4
$$

Prove that there exists a differentiable function $f ( x , y )$ defined in an open neighborhood N of $( 1 , 1 )$ in $\mathbf { R } ^ { 2 }$ such that $f ( 1 , 1 ) = 1$ and $( x , y , f ( x , y ) )$ lies in S for all $( x , y ) \in \mathcal { N }$

## PROBLEM 5

(A) Define uniform continuity for a function f defined on a metric space X with distance function $\rho ( x , y )$

B Prove that if $0 < \alpha < 1$ , then $F ( x ) = x ^ { \alpha }$ is uniformly continuous on $[ 0 , \infty )$

## PROBLEM 6

Let W be the subset of the space $C [ 0 , 1 ]$ of real-valued, continuous functions on [0, 1] satisfying the conditions:

$$
| f ( x ) - f ( y ) | < | x - y | \qquad \int _ { 0 } ^ { 1 } f ( x ) ^ { 2 } d x = 1
$$

(A) Prove that W is uniformly bounded, i.e., there exists $M > 0$ such that $| f ( x ) | \leq M$ for all $x \in [ 0 , 1 ]$ Hint: Show first that $| f ( 0 ) | \le 2$ for all $f \in W$

(B) Prove that W is a compact subset of $C [ 0 , 1 ]$ under the sup norm $\| f \| _ { \infty } = \operatorname* { s u p } _ { x \in [ 0 , 1 ] } | f ( x ) |$

## PROBLEM 7

A matrix T (with entries, say, in the field $\mathbf { C }$ of complex numbers) is diagonalizable if there exists a non-singular matrix S such that $S T S ^ { - 1 }$ is diagonal.
Prove that if $a , \lambda \in \mathbf { C }$ with $a \neq 0$ , then the following matrix is not diagonalizable:

$$
T = { \left( \begin{array} { l l l } { 1 } & { a } & { 0 } \\ { 0 } & { 1 } & { a } \\ { 0 } & { 0 } & { \lambda } \end{array} \right) }
$$

## PROBLEM 8

A linear transformation T is called orthogonal if it is nonsingular and ${ } ^ { t } T = T ^ { - 1 }$ . Prove that if $T : { \mathbf { R } } ^ { 2 n + 1 } \to { \mathbf { R } } ^ { 2 n + 1 }$ is orthogonal, then there exists a vector $\boldsymbol { v } \in \mathbf { R } ^ { 2 n + 1 }$ such that $T v = \pm v$

## PROBLEM 9

Let S be a real, n × n-symmetric matrix $S , { \mathrm { i . e . , ~ } } ^ { t } S = S$

(A) Prove that the eigenvalues of S are real.

(B) State and prove the Spectral Theorem for S.

## PROBLEM 10

Let Y be an arbitrary set of commuting matrices in $M _ { n } ( \mathbf { C } )$ $( \mathrm { i . e . , } A B = B A$ for all $A , B \in Y )$ Prove that there exists a non-zero vector $v \in \mathbf { C } ^ { n }$ which is a common eigenvector of all elements of Y.