"I really can't find a weak* closed-convex set that is a near-forced set, and thus get the Horsdave topology, and I believe that no mathematician in the world can prove that every *closed-convex set of x* is an approximate tight Chebyshev set. Shen Qi was not too depressed, but full of interest: "At first, I didn't realize that this was a dead end, because I didn't come to the end of a dead end. ”

"Do you have a good solution?" asked Müller.

Shen Qi already has the answer: "My solution is to establish a new topic separately for assertion (3), and I will tentatively name it 'Mill-Shen Near-Forcing Theorem', which needs to be further verified, and it can be regarded as a corollary of 'Mill-Shen Theorem'." ”

"Then the new 'Muller-Shen theorem' contains two assertions, (1) and (2), and this is enough, and we will complete the revision work in a few days, and publicly publish the results of the 'Muller-Shen theorem'. Shen Qi gave a solution.

"Actually...... We are equivalent to launching a 'reduced version' of the 'Muller-Shen theorem'. Professor Müller quickly understood Shen Qi's new plan.

Shen Qi nodded: "Professor Mueller's analogy is very appropriate, yes, the reduced version." In fact, the so-called reduced version is the standard version, because judgment (3) can be seen as a new product. ”

"It's actually two bodies of different lengths but roughly the same shape, with the same powertrain system, and then sold separately, well, Volkswagen does that a lot. "Professor Müller is indeed German, in a word.

"Haha!" Shen Qi laughed and said, "So I bought a Chevrolet." ”

As soon as Muller and Shen Qi combined, they did just that, first launching a low-profile version of the "Muller-Shen theorem" to meet the basic needs of the market and occupy this market segment first.

The high-end version of the "Muller-Shen theorem" will be launched in the near future, and when the time comes, it will be given a new name, and the "Muller-Shen theorem" is a good choice, so as to further enrich the product line and consolidate the market share and brand effect of "Muller-Shen" in this market segment.

The revision of the "Mill-Shen theorem" was quickly completed.

“...... After proof, we get the following theorem:

Let x be the Banach space, then the following argument is equivalent as:

(1) If x*∈s(x*) reaches its norm on s(x), then x* is the *concave point of the unit sphere b(x*);

(2) X is a strong smooth space. ”

Finally, the paper was checked, and Professor Müller personally uploaded the paper to arvix for pre-recording.

The task of officially submitting the paper was handed over to Shen Qi.

The title of the paper is "The Problem of Concave Spots and Strong Smooth Spaces in Banach Space", and both Mill and Shen Qi are the first authors.

The paper is 24 pages long and contains a complete proof of the "Mill-Shen theorem".

Of course, the "Mill-Shen theorem" was unilaterally claimed by Mill and Shen Qi, and it is up to the IMU to decide whether it can be included in the IMU's mathematical theorems.

Shen Qi thought that the quality of this paper was very high, and it belonged to the four major journals, so he submitted it to the Annals of Mathematics.

At the same time, the proof of the "Muller-Shen proximate theorem" was also carried out simultaneously, and this work was mainly completed by Shen Qi, because it was Shen Qi who proposed it.

It's another Wednesday in the blink of an eye.

This morning, Faltins habitually browsed Arvix and found a paper called "The Problem of Concave Points and Strongly Smooth Spaces in Barnach Spaces" by his colleague and a PhD student in the Department of Mathematics at the University.

"Oh, the ...... 'Mill-Shen theorem' thus proved applies to the approximation of a tight closed-convex subset, as well as the RNP properties of Banach spaces and martingale theory. It seems that Old Allen can finally publish a paper this year. Faltins read the paper in its entirety and felt that there was nothing wrong with it.

Grad Faltins, a German, rose to fame early, proving the sensational Moder conjecture in 1983, when Faltins was just 29 years old.

When he won the Fields Medal in 1986, Faltins was only 32 years old, and his wife was also a mathematician, and his teacher was the godfather of mathematics in the 20th century, Grothendieck.

The disciple of a famous teacher, married to a mathematician's wife, and the people around Faltings in his life were all related to mathematics.

As one of the top masters in mathematics today, Faltings is the editor-in-chief of the most authoritative mathematics journal, the Annals of Mathematics.

In the afternoon, the café on the third floor of the Mathematics Department Building, the usual coffee break.

Faltins took the coffee cup, walked over to the small table of Muller and Shen Qi, and sat down.

“ass?@@，*helu@* ful?。 Faltins spoke a word of German to Müller, and his expression seemed to be teasing.

"

Shen Qi was stunned: "Can the two professors speak English?"