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.
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.