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Here is the definition of a normed vector space my book uses:


And here is a remark I do not understand:


I do not understand that a sequence can converge to a vector in one norm, and not the other. For instance: Lets say $s_n$ converges to $u$ with the $\|\|_1$-norm. From definition 4.5.2 (i) we must have that $s_n$ becomes closer and closer to $u$. Why is it that it could fail in the other norm, when it can become as close as we want in the first norm?Are there any simple examples of this phenomenon?

PS:I know that they say we will see examples of this later in the book, but what comes later is too hard for me to udnerstand now.

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It should be mentioned that this never happens in finite dimensions. In finite dimensions, every norm is "equivalent" meaning that if a sequence converges in one norm, in converges in the other (which implies a series converges in one norm if it converges in the other). This should give you an idea of how bizarre things get when you have an infinite dimensional vector space. – Cameron Williams Mar 24 '14 at 0:24
up vote 4 down vote accepted

Consider the following sequence of elements in the space $V$ of finite sequences: $$ u_1=(1,0,0,\ldots),\ \ u_2=(0,\frac12,0,\ldots),\ \ u_3=(0,0,\frac13,0,\ldots) $$ Then $$ \sum_{k=1}^nu_k=(1,\frac12,\frac13,\ldots,\frac1n,0,\ldots) $$ Now consider these two norms on $u=(a_1,a_2,\ldots)$: $$ \|u\|_1=\sum_{k=1}^\infty|a_k|,\ \ \ \|u\|_2=\left(\sum_{k=1}^\infty|a_k|^2\right)^{1/2} $$ Then, for the $u_n$ defined above, $$ \left\|\sum_{k=M}^nu_k\right\|_1=\sum_{k=M}^N\frac1k,\ \ \ \left\|\sum_{k=M}^Nu_k\right\|_2=\sum_{k=M}^N\frac1{k^2} $$ So, in $\|\cdot\|_1$, the tails of the series $\sum_{k=1}^\infty u_k$ are unbounded, which means that the series diverges; while in $\|\cdot\|_2$, the tails go to zero, so the series converges.

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Thank you, so I guess in the first case we have that the distance between |(1,1/2,1/3...,1/n,0,0,...)-(1/1,1/2,1/3,1/4,1/5,1/6.......)|=|(0,0,0,0,0,0,0,1/‌​(n+1),1/(n+1),1/(n+3),.....| will never be as small as we want, but it will be that in the second case? Thank you for your help! – user119615 Mar 24 '14 at 1:26
Yes indeed. The point, as your book states, is that the notion of series depends on the choice of a topology, as the series is a limit of the sequence of partial sums. – Martin Argerami Mar 24 '14 at 4:16

You can get in trouble if the convergence is not absolute, i.e. $$\sum_n \|x_n\| = +\infty$$ but $$\lim_{N\to\infty} \sum_{n\le N} x_n $$ exists.

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This would be appropriate as a comment. – Sanath K. Devalapurkar Mar 24 '14 at 0:14
What is norm 1 and what is norm 2 in your example? – user119615 Mar 24 '14 at 0:17

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