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MMLU-Pro / 8844 / Let M be the set of bounded functions (i.e. …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Let M be the set of bounded functions (i.e. \sup_{x\in[a,b]}|f(x)|<\infty) in C[0,1]. Is the set F(x)=∫0xf(t)dt|f∈M{F(x)=\int_0^x f(t) dt | f \in M } a sequentially compact set? Answer 1 for yes and 0 for no. Furthermore, it can be proved using 1. Arzelà-Ascoli theorem, 2. Riesz representation theorem, 3. Banach fixed point theorem, 4. None of the above. Return the answers of the two questions in a list. For example, if you think the answer is no and Riesz representation theorem, then return [0,2].
Plain-text mathematical notation (without MathML)
Let M be the set of bounded functions (i.e. \sup_{x\in[a,b]}|f(x)|<\infty) in C[0,1]. Is the set F(x)=∫₀^(x)f(t)dt|f∈M a sequentially compact set? Answer 1 for yes and 0 for no. Furthermore, it can be proved using 1. Arzelà-Ascoli theorem, 2. Riesz representation theorem, 3. Banach fixed point theorem, 4. None of the above. Return the answers of the two questions in a list. For example, if you think the answer is no and Riesz representation theorem, then return [0,2].
Original LaTeX notation
Let M be the set of bounded functions (i.e. \sup_{x\in[a,b]}|f(x)|<\infty) in C[0,1]. Is the set ${F(x)=\int_0^x f(t) dt | f \in M }$ a sequentially compact set? Answer 1 for yes and 0 for no. Furthermore, it can be proved using 1. Arzelà-Ascoli theorem, 2. Riesz representation theorem, 3. Banach fixed point theorem, 4. None of the above. Return the answers of the two questions in a list. For example, if you think the answer is no and Riesz representation theorem, then return [0,2].

category

math

options

  1. [0, 2]
  2. [0, 3]
  3. [1, 3]
  4. [0, 1]
  5. [1, 1]
  6. [1, 0]
  7. [1, 2]
  8. [0, 4]
  9. [1, 4]
  10. [0, 0]

src

theoremQA-Math

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Source and history

upstream_id / Original problem ID
8844
task_id
689fe69e-5430-59d1-abbe-9af5ea4fa27c
task_key
default--test--8844
task_revision_id
1
config / split
default / test

Official source

initial import