## Math 508: Advanced Analysis

Homework 11
Lawrence Tyler Rush

<me@tylerlogic.com>

December 4, 2014

http://coursework.tylerlogic.com/courses/upenn/math508/homework11
### 1

Let L : X → Y be linear map between vector spaces X and Y such that x_{1},x_{2} ∈ X
are solutions, respectively, to
for some y_{1},y_{2} ∈ Y . Furthermore, let z≠0 be a solution to Lx = 0.

#### (a) Find a solution for Lx = 3y_{1}

The vector 3x_{1} is a solution since L(3x_{1}) = 3(Lx_{1}) = 3y_{1}.

#### (b) Find a solution for Lx = -5y_{2}

The vector -5x_{2} is a solution since L(-5x_{2}) = -5(Lx_{2}) = -5y_{2}.

#### (c) Find a solution for Lx = 3y_{1} - 5y_{2}

The vector 3x_{1} - 5x_{2} is a solution since L(3x_{1} - 5x_{2}) = 3(Lx_{1}) - 5(Lx_{2}) = 3y_{1} - 5y_{2}.

#### (d) Find a solution other than z and 0 for Lx = 0

The vector 2z is a solution since L(2z) = 2(Lz) = 2(0) = 0.

#### (e) Find two solutions of Lx = y_{1}

Both x_{1} + z and x_{1} + 2z are solutions since L(x_{1} + z) = Lx_{1} + Lz = y_{1} + 0 = y_{1}
and L(x_{1} + 2z) = Lx_{1} + 2Lz = y_{1} + 2(0) = y_{1}.

#### (f) Find another solution to Lx = 3y_{1} - 5y_{2}

The vector 3x_{1} - 5x_{2} + z is another solution since
L(3x_{1} - 5x_{2} + z) = 3(Lx_{1}) - 5(Lx_{2}) + Lz = 3y_{1} - 5y_{2} + 0 = 3y_{1} - 5y_{2}.

### 2

### 3

#### (a)

#### (b)

### 4

### 5

Since K(x,y) is continuous on [0,1] × [0,1] and [0,1] × [0,1] is compact in ℝ^{2}, then K
is uniformly continuous on [0,1] × [0,1]. Let ε > 0. Then for any x,z ∈ ℝ there exists a δ > 0 such that |x-z| < δ implies
|K(x,y) - K(z,y)| < ε for all y ∈ ℝ. This implies that
so that h is continuous.

### 6

If we rewrite
as αe^{x} where we define the constant α = ∫
_{0}^{1}e^{-y}u(y)dy we can simplify u(x) as

| (6.1) |

From this simplification we have e^{-x}u(x) = e^{-x}f(x) + λα which, after integrating both sides with respect to x over [0,1],
implies

Combining the last line of the above equation with equation 6.1 we end up with
Hence we have a unique solution for

so long as λ≠0.

### 7

Not worded properly.

### 8