I just now say “Hmm,this comment is wrong again”.After seeing your reply,It seems that my comment is not wrong again,because in your book is

,In spivak’s book is .

Now I understand,thanks for your reply!

In higher dimensions, the corresponding identification between matrices and linear transformations is dependent on the choice of basis, which is why it is important to keep the two concepts separate. But in one dimension the identification is basis-independent, and there is little harm in conflating the two concepts (or, for that matter, with identifying $1 \times 1$ matrices with scalars).

I think the definition of derivative in this book (and in Spivak’s “calculus on manifold”,etc.) is inconsistent.

When we say “A differentiable function has derivative at point “,we regard as a linear map from to $\mathbf{R}^n$.

But in the case of ,we regard as a real number.A real number is a real number,not a linear map,isn’t it?(Though a real number multiplying a stuff means a linear map )

It is very similar to such conditions,that is,the distinction between the matrix and the correspoinding linear map.Maybe somebody regard matrix as a linear map(I don’t know whether “somebody” exists or not,I just guess),but I regard matrix as a matrix, columns, rows.Only when a matrix multiplying a stuff forms a linear map…

But even though this method is wrong,I think I will be right only after some minor corrections.Instead of ,I need to construct a trigonometric function,this function is always non-negative,and at the place of ,this function should become large again.

Then let this function replace ,then I think that will be OK.

This is not an errata,but a personal advice.In lemma 16.4.6,you make use of Fejer kernel to construct an approximation to the identity.Why Fejer kernel?I think a long time to find the geometric meaning of the Fejer kernel,but the geometric meaning is rather complex,I make use of a sophiscated but wrong model of planet motion found by ancient astronomers to give Fejer kernel a geometric explanation(A circle A_2 moves around a circle A_1,and a circle A_3 moves around the circle A_2,etc…).

So I think the Fejer kernel approach is rather unintuitive to a begginer in Fourier analysis.Here is an intuitive approach:

Consider the function .When become larger and larger, will become thiner and thiner,and at last will become a approximation to the identity(In order to do this ,we have to multiply a constant to f_n(x),so that ).And it is easy to verify by using algebra of trigonometrics like ,we can change into trigonometric polynomials,that is done!

In Exercise 15.7.2,I think “Show that there exists a c>0 such that f(y) is non-zero whenever 0<|x-y|<c” should be “Show that there exists a c>0 such that f(y) is non-zero whenever 0<|x_0-y|<c”.

Sorry that my comment is not always right,so it needs your wisdom to judge which is right and which is wrong….

[Errata added, thanks - T.]