In Lang's "Real and Functional Analysis", he gives the following as the first problem of chapter XIV, which he says is due to Tate:
Let $E,F$ be Banach spaces, and let $f:E\rightarrow F$ be a map with the following property. There exists a number $C>0$ such that for all $x,y\in E$ we have $$|f(x+y)-f(x)-f(y)|\leq C$$ Show that there exists a unique linear map $g:E\rightarrow F$ such that $g-f$ is bounded in the sup norm. [Hint: Show that the limit $$g(x)=\lim_{n\rightarrow \infty} \frac{f(2^n x)}{2^n}$$ exists.]
I do not think this is correct as stated. My counterexample is $E=F=\mathbb{R}$ and $f$ such that $f(x+y)-f(x)-f(y)=0$. Such an $f$ satisfies Cauchy's functional equation, and the $g(x)$ derived as in the problem statement will be equal to $f$. However, with no regularity conditions on $f$, it is well known that it need not be linear. Hence the conclusion is false.
My questions are:
- Have I missed anything in my critique?
- What are the correct hypotheses for the theorem of Tate?
I would also appreciate a reference to Tate's work if possible, but I care about this less that the above points.
If you read the paper of Hyers (On the Stability of the Linear Functional Equation), he defines solutions of $f(x+y)=f(x)+f(y)$ to be linear functions. Using this definition clarifies what is being asked in the problem. However, the wording in Lang's book is indeed poor (unless I missed this definition somewhere).
Also, Lang has essentially the same problem in his algebra book (3rd ed., p.598), where it says that $g$ should be an additive function.