Sequence Spaces

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 \((u_n)\) and \((v_n)\) be two orthonormal sequences in a Hilbert space \(H\). Assume that

Show that if \((u_n)\) is a total orthonormal sequence, then \((v_n)\) is also a total orthonormal sequence. Hint: Assume for contradiction that \((v_n)\) is not total.

Consider the left shift operator \(L : \ell ^2 \to \ell ^2\), defined for a sequence \((x_n) \in \ell ^2\) by

1. Prove that \(\|L\| = 1\).
2. Explain why if \(\lambda \in \mathbb {C}\) with \(|\lambda | > 1\), then \(\lambda \in \rho (L)\), the resolvent set of \(L\).
3. Prove that that if \(\lambda \in \mathbb {C}\) with \(|\lambda | < 1\), then \(\lambda \in \sigma _p(L)\), the point spectrum of \(L\).
4. Deduce what is \(\sigma (L)\), the spectrum of \(L\).

Let \((u_n)\) and \((v_n)\) be two orthonormal sequences in a Hilbert space \(H\). Assume that \[ \sum _{n=1}^{\infty }\lVert u_n-v_n\rVert <1. \] Show that if \((u_n)\) is a total orthonormal sequence, then \((v_n)\) is also a total orthonormal sequence.

Hint: Assume for contradiction that \((v_n)\) is not total.

Let \((e_n)\) be a total orthonormal sequence in a Hilbert space \(H\). Let \((x_n)\subset H\) be a sequence. Prove that the two following statements are equivalent.

1. \(x_n\to x\) weakly.
2. The sequence \((x_n)\) is bounded and for any \(j\ge 1\), \(\langle x_n,e_j\rangle \to \langle x,e_j\rangle \) as \(n\to \infty \).

Hint: Show that if (ii) holds and \(y_N\in \operatorname {span}\{e_1,\ldots ,e_N\}\) then \(\langle x_n,y_N\rangle \to \langle x,y_N\rangle \) as \(n\to \infty \).

Let \((e_n)\) be an orthonormal sequence of a Hilbert space \(H\). Show that \(e_n\to 0\) weakly.

Let \(H\) be a Hilbert space. Recall that an operator \(T\in B(H)\) is Hilbert-Schmidt if there exits a complete orthonormal sequence, \((e_n)\in H\) such that \[ \sum _{n=1}^{\infty }\lVert Te_n\rVert ^2<\infty . \] Show that every Hilbert-Schmidt operator is compact.

If \((x_n)\) is a weakly convergent sequence in a normed space \(X\), say, \(x_n\to x_0\) weakly, show that there is a sequence \((y_n)\) of linear combinations of elements of \((x_n)\) which converges strongly to \(x_0\). Hint: First prove that \(x_0\in \overline {\operatorname {span}(x_n)}\).

Consider the operator \(T:\ell ^2\to \ell ^2\) defined for \(x=(\xi _j)\in \ell ^2\) by \[ Tx=(0,\xi _1\alpha _1,\xi _2\alpha _2,\ldots ), \] for some sequence \((\alpha _j)\) with \(\alpha _j\to 0\). Show that \(T\) is compact.

Let \(X\) and \(Y\) be normed vector spaces. Show that a linear operator \(T:X\to Y\) is compact if and only if for every sequence \((x_n)\) with \(\lVert x_n\rVert \le 1\), the sequence \((Tx_n)\) has a convergent subsequence.

Let \(X\) be a Banach space. Let \((f_n)\) be a sequence in \(X'\).

1. Show that \(f_n\to f\) weakly in \(X'\) implies that for all \(x\in X\), we have \(f_n(x)\to f(x)\).
2. If there is an \(f\in X'\) such that \(f_n(x)\to f(x)\) for all \(x\in X\), does \(f_n\to f\) weakly in \(X'\)? Explain your answer.

Let \((e_n)\) be an orthonormal sequence of a Hilbert space \(H\). Show that \(e_n\to 0\) weakly.

Let \((x_n)\) be a sequence in a Hilbert space \(H\). Show that: \[ x_n\to x\ \textbf {strongly} \quad \text {if and only if}\quad (x_n\to x\ \textbf {weakly}\text { and }\lVert x_n\rVert \to \lVert x\rVert ). \]

1. Define Cauchy sequence, and what it means for a linear space to be complete.
2. Let \(C[0,1]\) be the set of real-valued functions which are continuous on \([0,1]\). Show that \(C[0,1]\) is complete under under the uniform norm \(\lVert f\rVert =\sup _{x\in [0,1]}|f(x)|\).
3. Let \(f_n(x)=x^n\) for \(x\in [0,1]\). Find \(f=\lim _{n\to \infty }f_n\). Is \(f(x)\) continuous? Does this violate the completeness you showed in (a)? Explain.

Suppose \(f(z)\) is analytic in some neighborhood of the point \(z_0\) without the point \(z_0\) itself; let \(z_0\) be an essential singularity of \(f(z)\).

Show that for any complex number \(C\), there is a sequence of points \(z_n\) (\(n=0,1,2,\ldots \)) such that \[ z_n\to z_0\qquad \text {and}\qquad f(z_n)\to C. \] (In other words, in every neighborhood of an essential singularity, the function \(f(z)\) is arbitrary close to every complex number.)

Distributions. Let \(\Omega \subset \mathbb {R}^n\) be an open set.

1. Let \(\{f_n\}\subset C(\Omega )\), with \(f_n\) converging to \(f\in C(\Omega )\) pointwise, i.e., \(f_n(x)\to f(x)\) for all \(x\in \Omega \). Does it follow that \(f_n\) converges to \(f\) in the sense of distributions? Prove or give a counterexample.
2. Let \(\{\mu _n\}\subset \mathcal {D}'(\Omega )\) with \(\mu _n\) converging to \(\mu \in \mathcal {D}'(\Omega )\) in the sense of distributions. Let \(\alpha \) be a fixed multi-index. Does it follow that the derivatives \(D^\alpha \mu _n\) converge to \(D^\alpha \mu \) in the sense of distributions? Prove or give a counterexample.
3. Find a sequence of \(L^1(\mathbb {R})\) functions which do not converge in \(L^1(\mathbb {R})\) but which do converge in the sense of distributions.

Let \(\Omega \subset \mathbb {R}\) be open, and let \(C_0^\infty (\Omega )\) be the space of \(C^\infty \) functions with support in \(\Omega \). Define convergence of a sequence \(\varphi _n\) as follows: \(\varphi _n\to 0\) iff \(\exists \) a compact set \(K\subset \Omega \) such that

1. Support \(\varphi _n\subset K\) for all \(n\)
2. \(\displaystyle \lim _{n\to \infty }\sup _{x\in K}\lvert \varphi _n^{(p)}(x)\rvert =0\) for \(p\in Z^+\).

1. Define what is meant by a distribution on \(C_0^\infty (\Omega )\).
2. Define \[ \mu _f(\varphi )=\int _{-\infty }^{\infty } f(x)\bar {\varphi }(x)\,dx \] where \(f\in L^1_{\mathrm {loc}}(\Omega )\) and \(\varphi \in C_0^\infty (\Omega )\). Does this define a distribution on \(C_0^\infty (\Omega )\)? Give reasons.
3. Give a concrete example of a regular distribution and verify that is is a distribution.
4. Show how to define the derivatives of a regular distribution.

Show that there is a sequence of functions \(u_i\in S_\varphi \) so that \(u_i\to w_\varphi \) uniformly in \(\overline {B_r(\xi )}\).

Prove that an orthonormal sequence \(\{\phi _1,\phi _2,\ldots ,\phi _n\}\) in a Hilbert space \(\mathcal {H}\) is complete if and only if Parseval’s relation \[ \lVert f\rVert ^2=\sum _{n=1}^{\infty }|\langle f,\phi _n\rangle |^2 \] holds for all vectors \(f\in \mathcal {H}\).