Math 641-600 — Fall 2013

Assignment 1 - Due Thursday, September 5.

1. Let $V$ be a real finite dimensional inner product space, with inner product $\langle \cdot,\cdot \rangle_V$, and let $B=\{v_1,v_2,\ldots,v_n\}$ be an ordered basis for $V$.
   1. If $\Phi$ is the associated coordinate map, show that $\langle \cdot,\cdot \rangle_{\mathbb R^n} := \langle \Phi^{-1}(\cdot),\Phi^{-1}(\cdot) \rangle$ defines an inner product on $\mathbb R^n$.
   2. Show that if $\mathbf x, \mathbf y \in \mathbb R^n $, then $\langle \mathbf x,\mathbf y \rangle_{\mathbb R^n} = \mathbf y^TG\mathbf x$, where $G_{jk} = \langle v_k,v_j \rangle_V$. (The matrix is called the Gram matrix for $B$.)

2. In the previous problem, suppose that $B=\{v_1,v_2,\ldots,v_n\}$ is simply a subset of vectors in $V$ and that $U=\text{span}(B)$. Show that $B$ is a basis for $U$ (i.e. that $B$ is linearly independent) if and and only if $\mathbf y^TG\mathbf x$ is an inner product for $\mathbb R^n$.

3. Let U be a subspace of an inner product space V, with the inner product and norm being < ·,· > and ||·|| Also, let v ∈ V. (Do not assume that U is finite dimensional or use arguments requiring a basis.)
   1. Fix $v\in V$. Show that $p\in U$ satisfies $ \min_{u\in U}\|v-u\| = \|v-p\|$ if and only if $v-p$ is orthogonal to the subspace $U$.
   2. Show that $p$ is unique, given that it exists for $v$.
   3. Suppose $p$ exists for every $v\in V$. Since $p$ is uniquely determined by $v$, we may define a map $P: V \to U$ via $Pv:=p$. Show that $P$ is a linear map and that $P$ satisfies $P^2 = P$. ($P$ is called an orthogonal projection. The vector $p$ is the orthogonal projection of $v$ onto $U$.)

4. Let $V$, $B$, $U$ and $G$ be as in problem 1. ($B$ is assumed to be a basis for $U$.)
   1. Let $v\in V$ and $d_k =: \langle v,v_k\rangle_V$. Show that $p=\sum_j x_j v_j\in U$ is the orthogonal projection of $v$ onto $U$ if and only if the $x_j$'s satisfy the normal equations, $d_k = \sum_j G_{kj}x_j$.
   2. Show that the orthogonal projection $P:V\to U$ exists.
   3. Show that if B is orthonormal, then $Pv=\sum_j \langle v,v_j\rangle_V v_j$.

5. Equality holds in Schwarz's inequality if and only if $\{u,v\}$ is linearly dependent.

6. Suppose that $F\in C[0,1]$, $F(x)\ge 0$, and $F(x_0)>0$ for some $x_0\in [0,1]$. Show that there is a closed interval $[a,b]\subseteq [0,1]$, $a\ne b$, that contains $x_0$ and on which $F(x)\ge \frac12 F(x_0)$.

7. Let $B=\{v_1,\ldots,v_n\}$ be a basis for a vector space $V$. Define linear functionals $\varphi_k$, $k=1,\ldots, n$, via $\varphi_k(v_j)= \delta_{jk}$, where $\delta_{jk}$ is the Kronecker $\delta$.
   1. Show that $B^\ast := \{\varphi_1, \ldots, \varphi_n\}$ is a basis for $V^\ast$. ($B^*$ is called the dual basis for $B$.)
   2. Let $V=\mathbb R^n$ and suppose that $B\;=\;\{\mathbf x_1, \ldots,\mathbf x_n\}$ is a basis of column vectors for $\mathbb R^n$, and let $X=[\mathbf x_1 \cdots \mathbf x_n]$. Show that ${\mathbb R^n}^\ast$ may be identified with the set of $1\times n$ row vectors, and that $B^\ast$ is then the set of rows of $X^{-1}$.

Assignment 2 - Due Thursday, September 12.

1. This problem concerns several important inequalities.
   1. Show that if α, β are positive and α + β =1, then for all u,v ≥ 0 we have
      u^αv^β ≤ αu + βv.
   2. Let x,y ∈ Rⁿ, and let p > 1 and define q by q^-1 = 1 − p⁻¹. Prove Hölder's inequality,
      |∑_j x_jy_j| ≤ ||x||_p ||y||_q.
      Hint: Using the inequality in part (a). first prove it for ||x||_p = ||y||_q = 1. Scale to get the final inequality.
   3. Suppose $\varphi=[y_1 \ldots y_n] \in {\ell^{p}}^*$. Hölder's inequality implies that $\|\varphi\|_{\ell^{p*}}\le \|y\|_q$. Show that we actually have $\|\varphi\|_{\ell^{p*}} = \|y\|_q$.
   4. Let x,y ∈ Rⁿ, and let p > 1. Prove Minkowski's inequality,
      ||x+y||_p ≤ ||x||_p + ||y||_p.
      Use this to show that ||x||_p defines a norm on Rⁿ. Hint: you will need to use Hölder's inequality, along with a trick.

2. $L^2$ minimization. Find the straight line $y=a+bx$ that minimizes $\int_0^1 (e^{-x} - a - bx)^2dx$.
3. $L^1$ minimization. Find the straight line $y=a+bx$ that minimizes $\int_0^1 |e^{-x} - a - bx|dx$, by following these steps.
   1. Whatever the minimizer is, geometric considerations show that $e^{-x}$ and $a+bx$ will cross at two points, 0 < s < t < 1. Find these two points by minimizing, over $a,b$, the area $A$ between the $f(x)$ and $a+bx$: \[ A=\int_0^1 |e^{-x} - a - bx|dx = \int_0^s (e^{-x} - a - bx)dx + \int_s^t (a+ bx-e^{-x})dx +\int_t^1 (e^{-x} - a - bx)dx. \]
   2. Use the crossing conditions $a+bs=e^{-s}$ and $a+bt=e^{-t}$ to find $a$ and $b$.

4. Use your favorite software (mine is Matlab) and plot, on the same set of axes, $e^{-x}$ and the two minimization solutions found in the previous two problems.

5. Let V be a finite dimensional inner product space and let U be a subspace of V. Recall that the orthogonal complement of U is
   U^⊥ = {v ∈ V | < v,u > = 0 for all u ∈ U}
   Show that V = U⊕U^⊥, where ⊕ symbolizes the direct sum of vector spaces. Also, show that (U^⊥)^⊥ = U.

Assignment 3 - Due Thursday, September 19.

1. Suppose that $A$ is an $m\times n$ matrix, with $m>n$, so that the columns of $A$ are in $\mathbb R^m$. Assume that the rank of $A$ is $n$.
   1. Use the Gram-Schmidt to find a factorization of $A$ into the form $A=QR$, where $Q$ is an $m\times n$ matrix whose columns are orthonormal, and $R$ is an $n\times n$ upper triangular matrix.
   2. Show that $Q^TQ=I_{n\times n}$ and that the $QQ^T$ is the orthogonal projection of $\mathbb R^m$ onto the column space of $A$.

2. Let $m>n$. Suppose that $\mathbf b\in \mathbb R^m$ and that $A = [\mathbf a_1 \cdots \mathbf a_n]$ is as in the previous problem. We want to minimize $\| \mathbf b - \sum_{j=1}^n x_j\mathbf a_j\|_{\mathbb R^m}=\|\mathbf b - A\mathbf x\|_{\mathbb R^m}$ over $\mathbf x = [x_1 \ \cdots \ x_n]^T.\,$ Show that the minimizer is $\mathbf x_0 = R^{-1}Q^T\mathbf b$, where $A=QR$. (Hint: let $\mathbf z = R\mathbf x$, so that you are minimizing $\| \mathbf b - Q\mathbf z \|$ over $\mathbf z$. Then, use the normal equations.)

3. Fredholm Alternative. Let $V$ and $W$ be finite dimensional inner product spaces, with inner products $\langle \cdot,\cdot \rangle_{V}$ and $\langle \cdot,\cdot \rangle_{W}$, respectively. Also, let $L:V\to W$ be linear.
   1. Show that $\mathcal R(L) \subseteq \mathcal N(L^\ast)^\perp$, where $\mathcal R(L)$ is the range of $L$ and $\mathcal N(L^*)$ is the null space of the adjoint $L^\ast$.
   2. Show that $\mathcal R(L)= \mathcal N(L^\ast)^\perp$, by contradiction. If $\mathcal R(L) \ne \mathcal N(L^\ast)^\perp$, then there is a vector $w \ne 0$ such that $w\in \mathcal R(L)^\perp \cap \mathcal N(L^\ast)^\perp$. But, if $w\in \mathcal R(L)^\perp \cap \mathcal N(L^\ast)^\perp$, then $w=0$. (This means that $W=\mathcal R(L) \oplus \mathcal N(L^*)$.)

4. Suppose that $A$ is an $n\times n$ real matrix such that $\mathbf x^TA\mathbf x>0$ for $\mathbf x\ne \mathbf 0$. Use the Fredholm Alternative to determine whether $A$ is invertible.

5. Let U be a unitary, n×n matrix; that is, U*U = I. Do the following.
   1. Show that < Ux, Uy > = < x, y >.
   2. Show that the eigenvalues of U all lie on the unit circle, $|\lambda|=1$.
   3. Show that U is diagonalizable. (Hint: Modify the proof used in class to show that a self adjoint matrix is diagonalizable.)
   4. Suppose that U is real as well as unitary — i.e., orthogonal. For $n$ odd, show that 1 or − 1 is an eigenvalue of U. (It's possible to have both simultaneously.)

Assignment 4 - Due Thursday, September 26.

1. Let $A$ and $B$ be self-adjoint matrices, which may be real or complex. We say that $A\le B$ if and only if $\langle A\mathbf x,\mathbf x\rangle \le \langle B\mathbf x,\mathbf x\rangle$ for all $\mathbf x$.
   1. If $\lambda_1\ge \lambda_2,\ldots,\lambda_n$ are the eigenvalues of $A$ and $\tilde \lambda_1\ge \tilde \lambda_2,\ldots,\tilde \lambda_n$ are the eigenvalues of $B$, then show that $\lambda_k \le \tilde \lambda_k$.
   2. Show that $\text{Trace}(A) \le \text{Trace}(B)$ if $A\le B$.
   3. Show that if we increase a diagonal entry of $A$, then the resulting matrix $B$ satisfies $A\le B$.
   4. (Keener, problem 1.3(b)). Use the previous part to estimate the lowest eigenvalue of the matrix below. Keener gets $-\frac13$. Using matlab you get less than about $-2$. Can you beat $-\frac13$? \[ A=\begin{pmatrix}8 & 4 & 4\\ 4 & 8 & -4 \\ 4 & -4 & 3\end{pmatrix} \]
2. (This is a generalization of Keener's problem 1.3.5). Let $A$ be a self-adjoint matrix with eigenvalues $\lambda_1\ge \lambda_2,\ldots,\ge \lambda_n$. Show that for $ 2\le k < n$ we have \[ \max_U \sum_{j=1}^k \langle Au_j,u_j \rangle =\sum_{j=1}^k \lambda_j, \] where $U=\{u_1,\ldots,u_k\}$ is any o.n. set. (Hint: Put $A$ in diagonal form and use a judicious choice of $B$.)

3. Show that $\ell^\infty$ is a Banach space under the norm $\|\{x_j\}\|= \sup_j |x_j|$

4. Show that $\ell^2$ is a Hilbert space under the inner product \[ \langle \{x_j\},\{y_j\} \rangle :=\sum_{j=1}^\infty \bar y_j x_j. \]
5. Let $0\le \delta \le 1$. We define the modulus of continuity for $f\in C[0,1]$ by \[ \omega(f;\delta) := \sup_{|\,s-t\,|\,\le\, \delta}|f(s)-f(t)|, \ \text{where }\ s,t \in [0,1]. \]
   1. Explain why $\omega(f;\delta)$ exists for every $f\in C[0,1]$.
   2. Fix δ. Let S_δ = { ε > 0 | |f(t) − f(s)| < ε for all s,t ∈ [0,1], |s − t| ≤ δ}. In other words, for given δ, ε is in the set if |f(t) − f(s)| < ε holds for all |s − t| ≤ δ. Show that
      ω(f;δ) = inf S_δ
   3. Show that ω(f;δ) is non decreasing as a function of δ. (Or, more to the point, as δ ↓ 0, ω(f;δ) gets smaller.)
   4. Show that lim _δ↓0 ω(f;δ) = 0.

Assignment 5 - Due Thursday, October 3.

1. Calculus problem: Let g be C² on an interval [a,b]. Let h = b − a. Show that if g(a) = g(b) = 0, then
   ||g||_C[a,b] ≤ (h²/8) ||g′′||_C[a,b].
   Give an example that shows that 1/8 is the best possible constant.

2. Use the previous problem to show that if f ∈ C²[0,1], then the equally spaced linear spline interpolant f_n satisfies
   ||f − f_n||_C[0,1] ≤ (8n²) ^{− 1} ||f′′||_C[0,1]

3. Let $0<\alpha<1$ be fixed. Define $f(x) := x^\alpha$, $x\in [0,1]$. Show that $\omega(f;\delta) \le C\delta^\alpha$. where $C$ is indepedent of $\delta$.

4. Derive the trapezoidal rule for approximating $\int_0^1f(x)dx$, $f\in C[0,1]$, by integrating the linear spline (in $S^{\frac{1}{n}}(1,0)$) that interpolates $f$ at $x_j=\frac{j}{n}$, $j=0,\ldots,n$. Estimate the error involved.

5. Let $V$ be a Banach space. Suppose that there is an uncountable set of vectors $U\subset V$ with the property that if $u,v\in U$, then there exists $\varepsilon_0>0$ such that $\|u-v\|\ge \varepsilon_0>0$. Prove that $V$ is non separable. Use this to show that $L^\infty[0,1]$ is non separable.

6. Recall that the B-splines $N_m$ satisfy the recursion relation \[ N_m(x) = \frac{x}{m-1}N_{m-1}(x)+\frac{m-x}{m-1}N_{m-1}(x-1), \ m\ge 2. \] Use this to show that $N_3(x) = \frac12 \big((x)_+^2 - 3(x-1)_+^2 + 3(x-2)_+^2 - (x-3)_+^2 \big)$. Hint: $(x-a)(x-a)_+^k=(x-a)_+^{k+1}$.

Assignment 6 - Due Tuesday, October 15

1. Let y_j,  $j=1,\dots,n$. Consider the cubic-Hermite spline s(x) ∈ S^h(3,1), with h = 1/n, that satisfies s(j/n) = y_j and minimizes $\int_0^1(s'')^2dx$. Show that s(x) is actually in C⁽²⁾[0,1]; that is, show that s(x) ∈ S^h(3,2).

2. Variational/Finite-element problem. We want to solve the boundary value problem (BVP):  −u" = f(x), u(0) = u(1) = 0, f ∈ C[0,1].
   1. Let H be the set of all continuous functions vanishing at x = 0 and x = 1, and having L² derivatives. Also, let H have the inner product:
      ⟨f,g⟩_H = ∫₀¹ f ′(x) g ′(x) dx.
      Use integration by parts to convert the BVP into its "weak" form:
      ⟨u,v⟩_H = ∫₀¹ f(x) v(x) dx for all v ∈ H.

   2. Conversely, suppose that u ∈ H is also in C⁽²⁾[0,1] and that u satisfies
      ⟨u,v⟩_H = ∫₀¹ f(x) v(x) dx for all v ∈ H.
      Show that u satisfies the BVP.

   3. Let V = S^h(1,0), with h = 1/n. Thus V is spanned by φ_j(x) := N₂(nx-j+1), j = 1 ... n-1. (Here, N₂(x) is the linear B-spline.) Show that the least-squares approximation to u from V is y = ∑_j α_jφ_j(x) ∈ V, where the α_j's satisfy Gα = β, with
      β_j = ⟨ y,φ_j ⟩_H = ∫₀¹ f(x) φ_j(x) dx, j=1 ... n-1 and G_kj = ⟨ φ_j, φ_k ⟩_H.

   4. Show that G_kj = ⟨ φ_j, φ_k ⟩_H is given by
      G_j,j = 2n, j = 1 ... n-1
      G_j,j-1 = - n, j = 2 ... n-1
      G_j,j+1 = - n, j = 1 ... n-2
      G_j,k = 0, all other possible k.
      

3. Let $S=\{s\in C^{(2)}(\mathbb R)\,|\, s\ \text{is a cubic on}\ [j,j+1], j\in \mathbb Z\}$. (These are all cubic B-splines with knots at the integers and defined on all of $\mathbb R$.) Suppose $s\in S$ has compact support in $[0,M]$. Determine the smallest value of $M$ such that $s \not\equiv 0$.

4. Let $U:=\{u_j\}_{j=1}^\infty$ be an orthonormal set in a Hilbert space $\mathcal H$. Show that the two statements are equivalent. (You may use what we have proved for o.n. sets in general; for example, Bessel's inequality, minimization properties, etc.)
   1. $U$ is maximal in the sense that there is no non-zero vector in $\mathcal H$ that is orthogonal to $U$. (Equivalently, $U$ is not a proper subset of any other o.n. set in $\mathcal H$.)
   2. Every vector in $\mathcal H$ may be uniquely represented as the series $f=\sum_{j=1}^\infty \langle f, u_j\rangle u_j$.

5. Show that every separable Hilbert space has an o.n. basis.

Assignment 7 - Due Thursday, October 24

1. $f(x) = e^{x}$, $-\pi < x < \pi$.
   1. Find the complex form of the Fourier series for $f$.
   2. Sketch three periods of the $2\pi$-periodic function to which the series converges pointwise. (Hand-drawn is fine. No need to use a computer here.)
   3. Find the sum the series $\sum_{n=0}^\infty \frac{1}{n^2+1}$.
   4. Estimate the error $\|f-S_N\|_{L^2[-\pi,\pi]}$, where $S_N$ is the partial sum of the Fourier series for $f$.

2. Prove this: Let $g$ be a $2\pi$-periodic piecewise continuous function. Then, $\int_{-\pi+c}^{\pi+c} g(u)du$ is independent of $c$. (Remark: This holds for $g$ integrable on each bounded interval of $\mathbb R$.)

3. Use the previous result to show that if $f$ is $2\pi$-periodic and piecewise smooth, then it has the Fourier series $f(x) \sim a_0 + \sum_{n=1}^\infty a_n\cos(nx)+b_n\sin(nx)$, where \[ a_0=\frac{1}{2\pi} \int_0^{2\pi} f(x)dx, \ a_n=\frac{1}{\pi} \int_0^{2\pi} f(x)\cos(nx)dx, \ b_n = \frac{1}{\pi} \int_0^{2\pi} f(x)\sin(nx) dx, \] Formulate a theorem on the pointwise convergence of the series.

4. Find the Fourier series for $f(x) = x,\ 0 < x < 2\pi$. Sketch three periods of the $2\pi$-periodic function to which the series converges pointwise. (Hand-drawn is fine. No need to use a computer here.)

5. Find the Fourier series for $f(x) = \left\{ \begin{array}{cl} 1 & x\in [-\tfrac{1}{4} \pi , \tfrac{1}{4}\pi], \\ 0 & x \in (-\pi, \tfrac{1}{4}\pi) \ \text{or }x \in (\tfrac{1}{4}\pi , \pi). \end{array} \right. $

6. Consider the series $\sum_{n=-\infty}^\infty c_n e^{inx}$, where $\sum_{n=-\infty}^\infty |c_n| <\infty$. Show that the series $\sum_{n=-\infty}^\infty c_n e^{inx}$ converges uniformly to a $2\pi$-periodic continuous function $f(x)$ and the the series is the Fourier series for $f$. Also, show that the series converges to $f$ in $L^2[-\pi,\pi]$.

Assignment 8 - Due Thursday, November 14

1. A measurable function whose range consists of a finite number of values is a simple function. One can also define a simple function as a linear combination of a finite number of characteristic functions of measurable sets. Let $f(x)=x^2$, $-1\le x \le 2$. Find two simple functions $s_1$ and $s_2$ such that $s_1(x) \le f(x) \le s_2(x)$ and \[ \int_{-1}^2s_2(x)dx - \int_{-1}^2 s_1(x)dx < 0.01. \] How well do these integrals compare with $\int_{-1}^2 f(x)dx$?

2. Let F(s) = ∫ ₀^∞ e ^{− s t} f(t)dt be the Laplace transform of f ∈ L¹([0,∞)). Use the Lebesgue dominated convergence theorem to show that F is continuous from the right at s = 0. That is, show that
   lim _s↓0 F(s) = F(0) = ∫ ₀^∞f(t)dt

3. Let f_n(x) = n^3/2 x e^{-n x}, where x ∈ [0,1] and n = 1, 2, 3, ....
   1. Verify that the pointwise limit of f_n(x) is f(x) = 0.
   2. Show that ||f_n||_C[0,1] → ∞ as n → ∞, so that f_n does not converge uniformly to 0.
   3. Find a constant C such that for all n and x fixed f_n(x) ≤ C x^-1/2, x ∈ (0,1].
   4. Use the Lebesgue dominated convergence theorem to show that
      lim _n→∞ ∫ ₀¹ f_n(x)dx = 0.

4. Let L be a bounded linear operator on Hilbert space $\mathcal H$. Show that the two formulas for $\|L\|$ are equivalent:
   1. $\|L\| = \sup \{\|Lu\| : u \in {\mathcal H},\ \|u\| = 1\}$
   2. $\|L\| = \sup \{|\langle Lu,v\rangle| : u,v \in {\mathcal H},\ \|u\|=\|v\|=1\}$

5. Let $V$ be a Banach space and let $L:V\to V$ be linear. Show $L$ is bounded if and only if $L$ is continuous.

6. Consider the boundary value problem $-u''(x)=f(x)$, where $0\le x \le 1$, $\, f\in C[0,1]$, $\, u(0)=0$ and $u'(1)=0$.
   1. Verify that the solution is given by $u(x) = \int_0^1 k(x,y)f(y)dy$, where \[ k(x,y) = \left\{ \begin{array}{cl} y, & 0 \le y \le x, \\ x, & x \le y \le 1. \end{array} \right. \]
   2. Let $L$ be the integral operator $L\,f = \int_0^1 k(x,y)f(y)dy$. Show that $L:C[0,1]\to C[0,1]$ is bounded and that the norm $\|L\|_{C[0,1]\to C[0,1]}\le 1$. Actually, $\|L\|_{C[0,1]\to C[0,1]}=1/2$. Can you show this?

   3. Show that $k(x,y)$ is a Hilbert-Schmidt kernel and that $\|L\|_{L^2\to L^2} \le \sqrt{\frac{1}{6}}$.

Assignment 9 - Due Thursday, November 21

1. Finish the proof of the Projection Theorem: If for every $f\in \mathcal H$ there is a $p\in V$ such that $\|p-f\|=\min_{v\in V}\|v-f\|$ then $V$ is closed.
2. Prove this: If $L:\mathcal H\to \mathcal H$ is a bounded linear transformation, then $\overline{R(L)} = N(L^*)^\perp$.
3. Let $\mathcal H$ be a Hilbert space of functions that are defined on $[0,1]$. In addition, suppose that $\mathcal H \subset C[0,1]$, with $\|f\|_{C[0,1]} \le K\|f\|_{\mathcal H}$ for all $f\in \mathcal H$. (The Sobolev space $H^1$ has this property.)
   1. Show that the point-evaluation functional $\Phi_x(f) =f(x)$ is a bounded linear functional on $\mathcal H$.
   2. Let $x$ be fixed. Show that there is a kernel $k(x,y)\in \mathcal H$ such that \[ \Phi_x(f)=f(x) = \langle f,k(x,\cdot)\rangle \] (The kernel $k(x,y)$ is called a reproducing kernel and $\mathcal H$ is called a reproducing kernel Hilbert space.)
   3. For $x,z$ fixed, show that $k(z,x) = \langle k(z,\cdot),k(x,\cdot)\rangle$. In addition, let $\{x_j\}_{j=1}^n$ be any finite set of distinct points (i.e. $x_j\ne x_k$ if $j\ne k$) in $[0,1]$, show that the matrix $G_{jk} = k(x_k,x_j)$ is positive semidefinite; that is, $\sum_{j,k}c_k\overline{c_j}k(x_k,x_j)\ge 0$
   4. Suppose the matrix $G$ is positive definite and therefore invertible. Let $f\in \mathcal H$. Show that there are unique coefficients $\{c_j\}_{j=1}^n$ such that $s(x) =\sum_{j=1}^n k(x_j,x)c_j$ interpolates $f$ at the $x_j$'s.
4. Consider the finite rank (degenerate) kernel k(x,y) = φ₁(x)ψ₁(y) + φ₂(x)ψ₂(y), where φ₁ = 2x-1, φ₂ = x², ψ₁ = 1, ψ₂ = x. Let Ku= ∫₀¹ k(x,y)u(y)dy. Assume that L = I-λ K has closed range,

   Alermative set of functions: Keep φ₁, φ₂, and ψ₁ the same. For ψ₂, use ψ₂ = 4x − 3.

   1. For what values of λ does the integral equation
      u(x) - λ∫₀¹ k(x,y)u(y)dy =f(x)
      have a solution for all f ∈ L²[0,1]?
   2. For these values, find the solution u = (I − λK)⁻¹f — i.e., find the resolvent.
   3. For the values of λ for which the equation does not have a solution for all f, find a condition on f that guarantees a solution exists. Will the solution be unique?

5. Consider the Hilbert space $\ell^{\,2}$. Let $S=\{\{a_j\}_{j=1}^\infty \in \ell^{\,2}\colon \sum_{j=1}^\infty (1+j^2)\,|a_j|^2\le 1 \}$. Show that $S$ is a compact subset of $\ell^{\,2}$.

Updated 11/15/2013.