ChatGPT is one of the leading AI assistants developed by OpenAI Company.
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ChatGPT OpenAI
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Let $X$ be a topological vector space. All sets mentioned below are understood to be the subsets of $X$. Prove the following statement: If $A$ and $B$ are compact, so is $A + B$
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Prompt: Let $X$ be a topological vector space. All sets mentioned below are understood to be the subsets of $X$. Prove the following statement: If $A$ and $B$ are compact, so is $A + B$Truthfully, I am a little bit surprised by the math proof ability and reasoning ability of ChatGPT. To solve the topological math problem, ChatGPT firstly gives a outline of the proof and provides more details to support the proof. The detailed response is as follows: To prove that the sum of two compact sets in a topological vector space X, X is compact, we need to show that if A and B are compact subsets of X, then A+B is also compact. Proof: 1. Recall Definitions and Properties 2. Consider the Projection Maps 3. Use the Compactness of A and B 4. Image of a Compact Set Under a Continuous Map 5. Conclusion: Thus, we have shown that if A and B are compact subsets of a topological vector space X, then A+B is also compact.
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