陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「245B: Sample midterm questions」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Below the fold, I am giving some sample questions for the 245B midterm next week. These are drawn from my previous 245A and 245B exams (with some modifications), and the solutions can be found by searching my previous web pages for those courses. (The homework assignments are, of course, another good source of practice problems.) Note that the actual midterm questions are likely to be somewhat shorter than the ones provided here (this is particularly the case for those questi

(These questions are of course primarily intended for my students than for my regular blog readers; but anyone is welcome to comment if they wish.)

已知结果和反例

Question 1. Let be a -finite measure space, and let be a signed -finite measure. Show that the following are equivalent:

Question 2. Let be the real line with the Borel -algebra, and let be a -finite unsigned measure. Show that it is not possible for counting measure on the real line (restricted to ) to be absolutely continuous with respect to , i.e. .

证明或构造的主线

Question 3. Let , let be the Banach space of power summable real sequences with the usual norm. For each natural number n, let be the element of consisting of the sequence which equals 1 at the entry and 0 elsewhere, thus when and otherwise. Let be a sequence in a Banach space X.

Question 4. Let W be a vector space, let A be an index set, and for every , let be a subspace of W which is equipped with a norm . Suppose that for each , the normed vector space is a Banach space. Assume also the following compatibility condition: if and is a sequence in which converges in norm to x and in norm to y, then x is necessarily equal to y. Show that the space

阅读时建议盯住的点

be an increasing sequence of closed subspaces of H. Let be the closure of the union of these subspaces; this is another closed subspace of H. Show that for any x in H, the sequence converges in norm to , where is the orthogonal projection of x to V, and similarly for .

Question 6. Let H be a Hilbert space, and let V be a closed subspace of H which is non-trivial (i.e. not equal to ). Let be the orthogonal projection onto V.

值得单独记下的条目

• Show that there is at most one continuous linear transformation with the property that for all natural numbers n.
• If , show that there exists a continuous linear transformation with for all natural numbers n if and only if the sequence is bounded.
• If , show that the uniqueness claim 1. can fail.
• Show that has operator norm 1.
• Show that is self-adjoint, i.e. .
• Show that is idempotent , i.e. .
• Conversely, if is a self-adjoint idempotent continuous linear transformation of operator norm 1, show that for some non-trivial subspace of H.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源