Difference between revisions of "Convergence of a sequence"
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+ | {{Todo|preserve "interesting" example}} | ||
==Interesting examples== | ==Interesting examples== | ||
===<math>f_n(t)=t^n\rightarrow 0</math> in <math>\|\cdot\|_{L^1}</math>=== | ===<math>f_n(t)=t^n\rightarrow 0</math> in <math>\|\cdot\|_{L^1}</math>=== | ||
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This clearly <math>\rightarrow 0</math> - this is <math>0:[0,1]\rightarrow\mathbb{R}</math> which of course has [[Norm|norm]] {{M|0}}, we think of this from the sequence <math>(\|f_n-0\|_{L^1})^\infty_{n=1}\rightarrow 0\iff f_n\rightarrow 0</math> | This clearly <math>\rightarrow 0</math> - this is <math>0:[0,1]\rightarrow\mathbb{R}</math> which of course has [[Norm|norm]] {{M|0}}, we think of this from the sequence <math>(\|f_n-0\|_{L^1})^\infty_{n=1}\rightarrow 0\iff f_n\rightarrow 0</math> | ||
− | {{Definition|Real Analysis|Topology|Functional Analysis}} | + | {{Definition|Real Analysis|Topology|Functional Analysis|Metric Space}} |
Latest revision as of 13:30, 5 December 2015
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TODO: preserve "interesting" example
Interesting examples
fn(t)=tn→0 in ∥⋅∥L1
Using the ∥⋅∥L1 norm stated here for convenience: ∥f∥Lp=(∫10|f(x)|pdx)1p so ∥f∥L1=∫10|f(x)|dx
We see that ∥fn∥L1=∫10xndx=[1n+1xn+1]10=1n+1
This clearly →0 - this is 0:[0,1]→R which of course has norm 0, we think of this from the sequence (∥fn−0∥L1)∞n=1→0⟺fn→0