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Results tagged with real-analysis
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user 26522
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
9
votes
1
answer
493
views
Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$
It may be better to move this to a separate question.
Let me call a linear subspace $V \subset L^2(0,1)$ to be tame if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or …
- 13.8k
10
votes
1
answer
581
views
Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?
Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails …
- 13.8k
5
votes
0
answers
195
views
What are the possible $L^{\infty}$ closures of an integration-invariant linear subspace of $...
Let $S \subset C([0,1],\mathbb{R})$ be an $\mathbb{R}$-linear subspace that is invariant under the $T := \int_0^x$ integration operation: if $g \in S$ then the function $f = Tg$ defined pointwise by $ …
- 13.8k
12
votes
Why could Mertens not prove the prime number theorem?
Because the scale is too small in Mertens's theorem, and the prime number theorem as well as the Riemann hypothesis are hidden by the $O(1/\log{X})$ notation.
Indeed, the former amounts to strengthen …
- 13.8k