Useful notes! Kudos to professor Tao for writing them!

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If one has a sequence $latex {x_1, x_2, x_3, \ldots \in {\bf R}}&fg=000000$ of real numbers $latex {x_n}&fg=000000$, it is unambiguous what it means for that sequence to converge to a limit $latex {x \in {\bf R}}&fg=000000$: it means that for every $latex {\epsilon > 0}&fg=000000$, there exists an $latex {N}&fg=000000$ such that $latex {|x_n-x| \leq \epsilon}&fg=000000$ for all $latex {n > N}&fg=000000$. Similarly for a sequence $latex {z_1, z_2, z_3, \ldots \in {\bf C}}&fg=000000$ of complex numbers $latex {z_n}&fg=000000$ converging to a limit $latex {z \in {\bf C}}&fg=000000$.

More generally, if one has a sequence $latex {v_1, v_2, v_3, \ldots}&fg=000000$ of $latex {d}&fg=000000$-dimensional vectors $latex {v_n}&fg=000000$ in a real vector space $latex {{\bf R}^d}&fg=000000$ or complex vector space $latex {{\bf C}^d}&fg=000000$, it is also unambiguous what it means for that sequence to converge to a limit $latex {v \in {\bf R}^d}&fg=000000$ or $latex {v \in {\bf C}^d}&fg=000000$; it means…

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2 Comments

    1. I would say that he knows his math but I’m afraid that would be an understatement. 🙂

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