Dual Spaces and Bounded Linear Functionals

Consider the operator \(T : L^2[0, 1] \to L^2[0, 1]\) defined by

Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent set \(\rho (T)\).

Let \(a\) and \(b\) be non-zero (but possibly complex) elements in the complex Hilbert space \(\ell ^2\). Consider the bounded linear operator \(T : \ell ^2 \to \ell ^2\) defined by

1. Use the Fredholm Alternative Theorem to find a simple condition on \(a\) and \(b\) guar- anteeing that the equation

   \[(T - I)u = v\]

   has a unique solution \(u\) for all \(v \in \ell ^2\).

2. Find the Hilbert adjoint operator \(T^* : \ell ^2 \to \ell ^2\).

3. Given \(T : X \to Y\) a bounded linear operator, recall that the adjoint operator \(T^\times : Y'\to X'\) is defined

   \[(T^\times g)(x) = g(Tx).\]

   Find the adjoint operator \(T^\times : \ell ^2 \to \ell ^2\). Hint: Express \(g\) and \(T^\times g\) using the Riesz Representation Theorem.

4. Use the Fredholm Alternative Theorem to give conditions on \(v_0\) guaranteeing that

   \[(T - I)u = v_0\]

   has a unique solution \(u\). When it exists, find the solution.

Let \(X\) be a Banach space and \(A\), \(T\) be bounded linear operators from \(X\) to itself. Assume further that \(A\) is compact. Show that \(TA\) and \(AT\) are also compact operators.

For \(k \in \mathbb {N}\) let \(T_k : \ell ^1 \to \ell ^\infty \) be the linear operator given by

where \(e_j = (0, 0, \ldots , 1, \ldots , 0, 0, \ldots )\), i.e. it is the element of \(\ell ^\infty \) with a single 1 in the \(j\)th position and all other entries are zero.

1. Compute \(\|T_k\|\) for each \(k\), and conclude that each \(T_k\) is bounded.
2. Prove that the sequence \((T_n)\) is strongly operator convergent.
3. Prove that the sequence \((T_n)\) is not uniformly operator convergent.

Let \(H\) be a Hilbert space and \(A : H \to H\) be a bounded self-adjoint linear operator.

1. Prove that

   \[\|A\| = \sup _{x \neq 0} \frac {|\langle Ax, x\rangle |}{\|x\|^2}.\]
2. Prove that at least one of \(\|A\|\) or \(-\|A\|\) is an element of \(\sigma (A)\).

Let \(X\) be a normed space over \(\mathbb {R}\) and \(Y \subset X\) be a linear subspace. For \(x_0 \in X \setminus Y\) we define the linear functional

by

1. Prove that \(f\) is continuous if and only if \(x_0 \notin \overline {Y}\).

2. Prove that \(\overline {Y} = \bigcap \{\operatorname {Ker} g \mid g \in X', Y \subset \operatorname {Ker} g\}\), where \(X'\) is the dual of \(X\).

   Hint for part \((ii)\): The \(\subset \) direction is relatively easy. For the other direction, note that as given in part (i), \(f\) satisfies \(f|_Y = 0\) and \(f(x_0) = 1\). The Hahn-Banach theorem provides an extension of \(f\) to the entire space, use this to prove the other direction by contradiction.

Let \((\lambda _j)\) be a sequence that is dense in \([0, 1]\) and consider the linear operator \(T : \ell ^2 \to \ell ^2\) defined by its action on a sequence \((\xi _j) \in \ell ^2\) by

Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

Let \(X\) be a real Banach space and \(X^*\) denote its dual. Let \(C \subset X\) be a convex set containing the origin. Define

The goal of this problem is to prove that \(C^{**} = \overline {C}\).

1. Show that \(C^{**}\) is closed.
2. Show that \(\overline {C} \subset C^{**}\).
3. Show that \(\overline {C}\) is convex.

4. Assume there is \(x_0 \in X\) such that \(x_0 \in C^{**}\) but \(x_0 \notin \overline {C}\). Show there exists an \(f \in X^*\) such that for all \(x \in \overline {C}\) we have:

   \[f(x) < 1 < f(x_0).\]
5. Explain the contradiction in (d) and how it can be used to conclude with the desired result.

Let \(X\) be a Banach space over the complex numbers. Let \(X_{\mathbb {R}}\) be the vector space over the reals obtained by restricting the multiplication by scalars used to define \(X\) to the real line. Let \(f : X \to \mathbb {C}\) be a bounded linear functional on \(X\) and let \(\phi : X \to \mathbb {R}\) be its real part, defined by \(\phi (x) = \operatorname {Re} f(x)\) for all \(x \in X\).

1. Prove that \(\phi \) is a linear functional on \(X_{\mathbb {R}}\).

2. Prove that \(f\) can be reconstructed from \(\phi \) by:

   \[f(x) = \phi (x) - i\phi (ix), \quad \text {for } x \in X.\]

3. Show that \(f\) as a linear functional on \(X\) and \(\phi \) as a linear functional on \(X_{\mathbb {R}}\) have the same norm, i.e.

   \[\|f\|_{X^*} = \|\phi \|_{X_{\mathbb {R}}^*},\]

   where \(X^*\) is the dual of \(X\) and \(X_{\mathbb {R}}^*\) is the dual of \(X_{\mathbb {R}}\). Hint: To prove one of the inequalities, it may be helpful to find \(\lambda \in \mathbb {C}\) such that \(|f(x)| = f(x/\lambda )\) for a fixed \(x\) with \(f(x) \neq 0\).

Let \((\lambda _j)\) be a sequence that is dense in \([0, 1]\) and consider the linear operator \(T : \ell ^2 \to \ell ^2\) defined by its action on a sequence \((\xi _j) \in \ell ^2\) by

Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence of finite rank operators \((T_k)\) that converges to \(T\) in an appropriate sense.

Let \(X\) be a Banach space. Let \(y \in X\) be fixed and let \(f : X \to \mathbb {R}\) be a bounded linear functional. Show that there is a constant \(C > 0\) such that for all scalars \(\alpha \) with \(|\alpha | < C\) the equation

has a unique solution on the open ball \(\{x \in X : \|x - y\| < 1\}\).

Let \(X\) be a real Banach space. Assume \(f \in X^*\) has a closed nullspace \(\mathcal {N}(f)\). The goal of this problem is to show that \(f\) must be a bounded linear functional.

1. Let \(x_0 \in X\) be such that \(f(x_0) = 1\). Explain why there is an \(\epsilon > 0\) for which the ball

   \[B(x_0, \epsilon ) \equiv \{x \in X \mid \|x - x_0\| < \epsilon \}\]

   satisfies \(B(x_0, \epsilon ) \subset X - \mathcal {N}(f)\).

2. Prove that \(f(x) > 0\) for all \(x \in B(x_0, \epsilon )\), where \(\epsilon \) is as in part (a). To prove this, you may assume for contradiction that there is some \(y \in B(x_0, \epsilon )\) with \(f(y) \le 0\).
3. Any \(x \in B(x_0, \epsilon )\) can be written as \(x = x_0 + \epsilon u\) for some \(u\) with \(\|u\| < 1\). Use the result of part (b) to show that \(|f(u)| < 1/\epsilon \).
4. To conclude, give an upper bound for \(\|f\|\).

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence \((T_k)\) of finite rank operators that converges to \(T\) in an appropriate sense.

Let \(T : H \to H\) be a bounded linear operator on a Hilbert space \(H\). Show that \(T\) is compact if and only if \(T^*T\) is compact.

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence \((T_k)\) of finite rank operators that converges to \(T\) in an appropriate sense.

Let \(H\) be a Hilbert space endowed with the inner product \(\langle \cdot , \cdot \rangle \). Let \(T\) be a bounded linear operator satisfying \(\langle T x, x\rangle \ge 0\) for all \(x \in H\). Show that \(I + T\) is injective.

Let \(T : H \to H\) be a bounded linear operator on a Hilbert space \(H\). Let \(A\) and \(B\) be subsets of \(H\) such that \(T(A) \subset B\). Show that \(A^\perp \supset T^*(B^\perp )\). Note: We used the notation \(T(A) = \{T x \mid x \in A\}\).

Let \(T : X \to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that \(\sigma (T)\) lies in the complex plane disk: \(\{\lambda \in \mathbb {C} \mid |\lambda | \le \|T\|\}\).

Let \(X\) be a real Banach space, \(f:X\to \mathbb {R}\) be a bounded linear functional and \(T:X\to X\) a bounded linear operator. In the following \(y\in X\) is fixed and \(\alpha \in \mathbb {R}\).

1. Prove that there exists a constant \(C>0\) such that if \(\lvert \alpha \rvert <C\), then the non-linear equation \[ x+\alpha f(x)Tx=y, \tag {1} \] has a unique solution \(x\) in the ball \(B=\{x\in X\mid \lVert x-y\rVert \le 1\}\).
2. Suggest an iterative procedure for approximating the unique solution \(x\) to equation (1).

Consider the complex function \(f(z)=\ln ((z-1)(z-2))\). Find the branch points. Choose a branch cut and find the corresponding principal branch of the function.

You must justify the choice of angles in the dual polar coordinate representation.

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that the spectrum \(\sigma (T)\) lies in the complex plane disk: \[ \{\lambda \in \mathbb {C}:\lvert \lambda \rvert \le \lVert T\rVert \}. \]

Let \(T\) be a bounded linear operator on a complex Hilbert space \(H\). Show that the operator \(I+T^*T\) is injective.

Consider the operator \(T:L^2[0,1]\to L^2[0,1]\) defined by \((Tx)(t)=tx(t)\).

1. Show that \(T\) is self-adjoint and satisfies \(\langle x,Tx\rangle \ge 0\) for every \(x\in L^2[0,1]\).
2. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent \(\rho (T)\).

Consider the multiplication operator \(T:L^2[0,1]\to L^2[0,1]\) defined by \(Tx(t)=tx(t)\).

1. Show that \(T\) is a bounded, self-adjoint linear operator.
2. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuous spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\), and resolvent \(\rho (T)\).
3. Show that the corresponding spectral family \((E_\lambda )_{\lambda \in \mathbb {R}}\) is given by \[ E_\lambda x= \begin {cases} 0, & \lambda <0,\\ v_\lambda , & 0\le \lambda \le 1,\\ x, & \lambda >1, \end {cases} \qquad \text {where}\qquad v_\lambda (t)= \begin {cases} x(t), & 0\le t\le \lambda ,\\ 0, & \lambda <t\le 1. \end {cases} \]

Let \(X\) be a real Banach space. Assume \(f\in X^*\) has a closed nullspace \(\mathcal {N}(f)\). The goal of this problem is to show that \(f\) must be a bounded linear functional.

1. Let \(x_0\in X\) be such that \(f(x_0)=1\). Explain why there is an \(\epsilon >0\) for which the ball \[ B(x_0,\epsilon )\equiv \{x\in X\mid \lVert x-x_0\rVert <\epsilon \} \] satisfies \(B(x_0,\epsilon )\subset X-\mathcal {N}(f)\).
2. Prove that \(f(x)>0\) for all \(x\in B(x_0,\epsilon )\), where \(\epsilon \) is as in part (a). To prove this, you may assume for contradiction that there is some \(y\in B(x_0,\epsilon )\) with \(f(y)<0\).
3. Any \(x\in B(x_0,\epsilon )\) can be written as \(x=x_0+\epsilon u\) for some \(u\) with \(\lVert u\rVert <1\). Use the result of part (b) to show that \(\lvert f(u)\rvert <1/\epsilon \).
4. To conclude, give an upper bound for \(\lVert f\rVert \).

Let \(T:X\to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that \(\sigma (T)\) lies in the complex plane disk: \[ \{\lambda \in \mathbb {C}\mid \lvert \lambda \rvert \le \lVert T\rVert \}. \]

Let \(T\) be a bounded linear operator on a complex Hilbert space \(H\). Show that the operator \(I+T^*T\) is injective.

Let \(X\) be a Banach space, \(f:X\to \mathbb {R}\) be a bounded linear functional, \(y\in X\) fixed and \(\alpha \in \mathbb {R}\).

1. Prove that there exists a constant \(C>0\) such that if \(\lvert \alpha \rvert <C\), then the non-linear equation \[ x+\alpha f(x)x=y, \tag {2} \] has a unique solution \(x\) in the ball \(B=\{x\in X\mid \lVert x-y\rVert \le 1\}\).
2. Suggest an iterative procedure for approximating the unique solution \(x\) to equation (2).

Let \((\lambda _j)\) be a sequence real numbers with \(\lambda _j\ne 1\) for all \(j\) and \(\lambda _j\to 1\). Consider the operator \(T:\ell ^2\to \ell ^2\) defined for \((\xi _j)\in \ell ^2\) by \[ T(\xi _j)=(\lambda _j\xi _j). \]

1. Find the spectrum \(\sigma (T)\), point spectrum \(\sigma _p(T)\), continuum spectrum \(\sigma _c(T)\), residual spectrum \(\sigma _r(T)\) and resolvent set \(\rho (T)\).
2. Give a condition on the \(\lambda _j\) for \(T\) to be invertible.

Let \(T:L^1[0,1]\to L^2[0,1]\) be a bounded linear operator. Let \(a\in L^\infty [0,1]\) and \(g\in L^1[0,1]\).

1. Prove that there exists a constant \(C>0\) such that if \(\lVert a\rVert _{L^\infty }<C\) then the equation \[ f+a(Tf)^2=g \tag {1} \] has a unique solution \(f\) in the set \[ B=\{u\in L^1[0,1]:\lVert u-g\rVert _{L^1}\le 1\}. \] Note: the constant \(C\) may depend on \(T\) and/or \(g\).
2. Suggest an iterative procedure for approximating the unique solution \(f\) to equation (1).

Let \(X\) be a separable Hilbert space and \((e_n)\) be a total orthonormal sequence of \(X\). Let \(T:X\to X\) be a bounded linear operator satisfying \[ \sum _{n=1}^{\infty }\lVert Te_n\rVert ^2<\infty . \tag {2} \] Define the operators \(T_k:X\to X\), \(k=1,2,\ldots \), that applied to an arbitrary \[ x=\sum _{n=1}^{\infty }x_ne_n\in X \qquad \text {give}\qquad T_kx=\sum _{n=1}^k x_nTe_n. \]

1. Are the operators \(T_k\) compact? Explain why or why not.
2. Using equation (2), show that the sequence \((T_k)\) converges uniformly to \(T\), i.e. show that \(\lVert T-T_k\rVert \to 0\) as \(k\to \infty \).
3. Is \(T\) compact? Explain why or why not.

Let \(\ell ^{1'}\) be the dual space of \(\ell ^1\). The objective here is to show that \(\ell ^{1'}\) is isomorphic to \(\ell ^\infty \). You may use that a Schauder basis for \(\ell ^1\) (and \(\ell ^\infty \)) is \((e_k)\), where \(e_k=(\delta _{kj})\), i.e. the sequence with zeroes everywhere except for a \(1\) in the \(k\)–th position.

1. Show that any \(f\in \ell ^{1'}\) corresponds to a sequence \(g=(\gamma _k)\in \ell ^\infty \).
2. Show that any sequence \(g=(\gamma _k)\in \ell ^\infty \) corresponds to some \(f\in \ell ^{1'}\).
3. Parts (a) and (b) define a bijective linear mapping \(G:\ell ^{1'}\to \ell ^\infty \). Prove that \(G\) is an isomorphism by showing \(\lVert f\rVert _{\ell ^{1'}}=\lVert G(f)\rVert _{\ell ^\infty }\) for any \(f\in \ell ^{1'}\).

Let \(K:L^2[0,1]\to L^2[0,1]\) be the linear operator defined by \[ (Kx)(t)=\int _0^1 k(t,\tau )x(\tau )\,d\tau \] where \(k:[0,1]\times [0,1]\to \mathbb {R}\) is continuous.

1. Show that \(K\) is bounded and give an upper bound for \(\lVert K\rVert \) (in terms of \(k\)).
2. Show that there is a constant \(C\) such that if \(\lvert \mu \rvert <C\), the operator \(I+\mu K\) is invertible.
3. Assuming \(\lvert \mu \rvert <C\), where \(C\) is as in part (b), express \((I+\mu K)^{-1}\) as a series. Specify whether the series converges and in what sense.

Let \(A\) be an \(n\times n\) real matrix. Consider the linear initial value problem \[ \begin {cases} \dfrac {dx}{dt}=Ax,\\ x(0)=x_0. \end {cases} \] Give an algebraic condition on the matrix \(A\) which holds if and only if all solutions of the initial value problem are bounded for all \(t\in \mathbb {R}\). Prove the equivalence.

Suppose that \(E\) is a bounded linear operator on a Hilbert space \(H\), with operator norm \(\lVert E\rVert =r<1\). Show that the operator \(I-E\), where \(I\) is the identity, is bounded and invertible. Give an estimate for \(\lVert (I-E)^{-1}\rVert \), and a convergent series representation for \((I-E)^{-1}\).

1. Let \(G\) be an open subset of \(\mathbb {R}^N\) and let \(f\in L^2(G)\). Show that \(f\) defines a continuous linear functional on \(H^1(G)\).
2. State the Lax-Milgram theorem for subspaces \(V\subset H^1(G)\).

3. Show that the Lax-Milgram theorem may be applied to the form \[ a(u,v)=\sum _{i=1}^{N}\int _G \partial _i u\,\partial _i\overline {v} +\int _G u\overline {v} \] and \(f\) as above.

   Assume that \(G\) is a bounded open set with “very smooth” boundary and let \(u\) be the Lax-Milgram solution just determined. What partial differential equation (in the sense of distributions) will be solved by \(u\) and what boundary conditions does \(u\) satisfy (please give an argument to prove the latter)?

1. State the Riesz Representation Theorem (a proof is not required).
2. Use the Riesz Representation theorem to prove that a bounded linear operator on a Hilbert Space has an adjoint. Prove that the adjoint is a bounded linear operator on the Hilbert space.
3. Find the adjoint for the operator \[ (Ku)(y)=\int _0^1 \min (x,y)u(x)\,dx. \]

Problem. In an isolated region of the Canadian Northwest Territories, a population of white wolves, \(x\), and one of silver foxes, \(y\), compete for survival (for each population one unit represents 100 individuals). They have a common, limited food supply, which consists mainly of mice. A mathematical biologist models the dynamics of the competing species by the nonlinear system \[ \begin {aligned} x' &= x-x^2-xy,\\ y' &= \frac {3}{4}y-y^2-\frac {1}{2}xy. \end {aligned} \] Can the two species survive together according to this model?

Prove that if \(L\) is a bounded linear operator in a Hilbert Space \(H\), then the adjoint operator \(L^*\) exists.

Consider the space of real-valued functions \(H^2(0,1)\) and define the quadratic form \[ a(u,v)=\int _0^1 u''v''\,dx+b\int _0^1 u'v'\,dx+c\int _0^1 uv\,dx, \] where \(b\) and \(c\) are positive constants. Show that \(a\) is a form to which the Lax-Milgram theorem may be applied and state the result thus obtained. Let \(f\in L^2(0,1)\), and restrict the form \(a\) to the space \(H^2_0(0,1)\). Show that \(f\) may be thought of as a continuous linear functional on \(H^2_0(0,1)\), and apply the Lax-Milgram theorem to the pair \(a,f\). Determine a boundary value problem which is solved in a distributional sense by the Lax-Milgram solution and show that this solution is at least three times continuously differentiable.

1. State the Riesz Representation Theorem.
2. Use the Riesz Representation theorem to prove that a bounded linear operator on a Hilbert Space has an adjoint. Prove that the adjoint is a bounded linear operator on the Hilbert space.
3. Find the adjoint for the operator \[ (Ku)(y)=\int _0^1 \min (x,y)u(x)\,dx. \]

Let \(G\) be an open subset of \(\mathbb {R}^N\) and let \(f\in L^2(G)\). Show that \(f\) defines a continuous linear functional on \(H^1(G)\).

Let \(H\) be a Hilbert space and \(L:H\to H\).

1. Find and prove the condition that ensures \[ Lu=\int _0^1 k(x,y)u(y)\,dy \] is a bounded linear operator in \(L^2[0,1]\).
2. Under what condition does the adjoint operator \(L^*\) exist? Assuming that this condition is satisfied, give \(L^*\).

Derive the boundary value problem for a minimizer \(w_0\) of the functional \[ \min _{\substack {w\in C^2[0,1]\\ w(0)=0}} \frac {\int _0^1\left (\phi (x)(w'')^2+w^2\right )\,dx} {\int _0^1\phi (x)(w')^2\,dx+\beta w(1)^2} \] where \(\phi (x)>0\), \(\forall x\in [0,1]\), \(\beta \) is a real parameter.

A bounded linear operator \(P\) in a complex Hilbert space \(\mathcal {H}\) is a projection if and only if \(P^2=P\). Prove that a projection \(P\) is an orthogonal projection (so that \(P\mathcal {H}\perp (1-P)\mathcal {H}\)) if and only if \(P\) is selfadjoint. (You may use the known theorem that a bounded linear operator \(A\) in \(\mathcal {H}\) is selfadjoint if and only if \((Ax,x)\) is real for all \(x\in \mathcal {H}\).)