Corrections to my book “A First Course in Functional Analysis”

In this page I list corrections (or helpful remarks) that were found in my book “A First Course in Functional Analysis“.  I wish to thank whoever found these mistakes and was kind enough to inform me (credits given below in initials).

A revised version with the following corrections and several other minor improvements is available – just ask me. MOST CORRECTIONS ARE FIXED IN THE SECOND PRINTING.

Corrections for the first version of the book (2017):

  1. p. 34, lines 10,11: “ordered” should be “partially ordered” (RP)
  2. p. 49, Exercise 4.1.9: for the integration by parts formula to hold, one needs to assume also that f is continuous. (RP)
  3. p. 54, line 6 from bottom: f(0) \neq f(1) should be f(0^+) \neq f(1^-). (RP)
  4. p. 57, Exercise 4.5.8: “for every integral” should be “for every interval“.
  5. p. 90, Equation (6.10): one should apply complex conjugation to the kernel function in the integral formula. This correction is not fixed in the second printing. (Adi Levy).
  6. p. 100, Exercise 6.3.21: “Theorem 6.3.17 theorem“.
  7. p. 115, Exercise 7.5.7: the dual of X + Y should be X* + Y* (with the appropriate q norm).
  8. p. 126: In the first few lines of the proof (top of page), “exists a B \in B(H,K)” should be replaced by “exists a linear map B from H to K”. Explanation: we say that it suffices to find B in B(H,K) that satisfies the equation on line 4. However, by Theorem 8.3.4, it really suffices to to find a linear B from H to K, not necessarily bounded; indeed the B we found is defined to be the inverse of A*, which is not known to be bouned yet (this is the whole point!). (RP)
  9. p. 139 in the statement of Lemma 9.2.1 the infimum in the displayed equation should be greater or equal than 1/2. It is not hard to see 1/2 can be replaced by any 0<r<1, and hence “greater or equal than” can be replaced by strictly greater than, but what we prove is the weak inequality. Consequently, whenever this lemma is invoked further down the section it should be greater or equal than 1/2.
  10. p. 142 in the poof of Theorem 9.2.12 Y_{n+1} should be Y_{n-1}.
  11. p. 148 the penultimate line of the proof of Lemma 10.2.7: lambda should be |lambda| (missing absolute value). (RP)
  12. p. 154, first paragraph of proof of Theorem 10.4.4: f(x) = sqrt(x) should be g(x) = sqrt(x). (RP)
  13. p. 162 equation (11.4): df should be dt.
  14. p. 163 line 9 (in item 6 of Theorem 11.1.1), as well as p. 169 lines 9 and 10: the operator A should be K.
  15. p. 188, line 8 from bottom: [-R,R] \times \bR should be  \bR \times [-R,R] (the domain of integration according to the (t,w) variables is finite in the w direction).
  16. p. 229, Corollary A.3.24: The corollary says that every continuous function from a compact space X into R attains a maximum and a minimum. The corollary is false if X is empty, because then the empty function is a (continuous!) function from X into R, but it doesn’t attain any value. The statement can be corrected by adding the assumption that X is not empty. Some authors don’t consider the empty set to be a topological space, and exclude this case right from the definition of a topological space. Had I followed that practice, the statement would have been true as is. (DR)

Remarks:

  1. p. 7 In the proof of the Stone-Weierstrass theorem: the neighborhoods U_y and V_x should be open neighborhoods (this is meant implicitly, though there is common usage of the word “neighborhood of x” that means: “set that contains an open set containing x”. To apply the finite covering property directly , one needs the cover to consist of open sets. (RP)
  2. On p. 40 – the statement of Theorem 3.5.1 is not made in a precise way. It is stated that if one has a sequence v_1, v_2, … of vectors in an inner product space G, then there exists an orthonormal sequence e_1, e_2, … with the same span. The problem is that if v_n = 0 for all n, then the only orthonormal set with the same span is the empty set, which cannot be written as e_1, e_2  (unless one flexes the imagination a bit). One way to fix this is simply to assume that at least one of the v_i’s is not zero. One can also fix the statement by saying that “given a finite or countable set, there exists a finite or countable orthonormal set with the same span”, which is correct if one interprets the empty set as a basis (even an orthonormal basis, in a vacuous sense) for the zero space {0} – this depends on whether you like the convention that an empty linear combination gives 0. The statement about a linearly independent sequence v_1, v_2, … holds as it is written. Note also that Corollary 3.5.2 (which says that every separable Hilbert space has an orthonormal basis) holds true as written, since the empty set is an orthonormal basis for the zero space G = {0}. (DR)
  3. p. 140 Lemma 9.2.6: The assertion that (I-A)Y is closed for every closed subspace Y holds true without the assumption that the kernel is trivial. The proof as it is does not cover the general case. After pointing this out I leave the simple argument (similar to arguments in this section) to the reader. The revised version will contain an improved version of this lemma. (RP)
  4. p. 154 Theorem 10.4.2 item 4: In order to invoke the Stone-Weierstrass theorem as we do in the proof, the sequence g_n of polynomials should be understood as polynomials in z and “z bar”, not just in z. This is not explained well in the second printing, too. It is worth noting that, by a Theorem of Lavrentieff, it is true that a continuous function on the spectrum of a compact operator (or more generally, any compact subset of the plane with a connected complement and an empty interior) can be uniformly approximated by polynomials just in z, but this requires much more than the SW theorem.