# Tagged Questions

Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

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### Example of a topological vector space such that $E = M \oplus N$ algebraically, but not topologically

I have the following question: Give an example of a topological vector space $E$ with subspace $M$ and $N$, such that $E = M \oplus N$ algebraically, but not topologically (so $E \ncong M \sqcup N$...
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### Prove or disprove $2abc(a+b+c)\ge 3(a^2b^2c^2+1)$

Let $a,b,c>0,ab+bc+ca=3$, prove or disprove $$2abc(a+b+c)\ge 3(a^2b^2c^2+1)$$ Now I can't find any counterexample
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### A concave positive function on $[1,\infty)$ is uniformly continuous
Let $f$ be a concave positive function on $[1,\infty)$, then $f$ is uniformly continuous on $[1,\infty)$. This was a true or false problem that I couldn't prove to be true, so I'm thinking that maybe ...
### Assume that the function $f(x)$ is continuous and $\lim_{n\to\infty}f_n(x)=f(x)$. Does this imply that $f_n(x)$ is uniform convergent?
I wonder if this is true: Let $(f_n)$ be a sequence of real-valued functions defined on a set $S\subset\mathbb{R}$. Assume that the function $f(x)$ is continuous and $\lim_{n\to\infty}f_n(x)=f(x)$. ...