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Uniform convergence, functions,sequences,subsequences

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#real-analysis #limits #uniform-convergence #cauchy-sequences #sequence-of-function
120
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Examples of spaces where compact convergence is not equivalent to local uniform convergence.

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#general-topology #compactness #examples-counterexamples #uniform-convergence #sequence-of-function
153
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check uniformly convergence of $f_n(x)=\begin{cases} nx & \text{ if } 0\leq x \leq \frac{1}{n}\\1& \text{ if } \frac{1}{n}<x\leq 1\\ \end{cases}$

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#real-analysis #solution-verification #uniform-convergence #sequence-of-function
60
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Convergence in sequential Lebesgue spaces.

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#analysis #measure-theory #lebesgue-integral #lp-spaces #sequence-of-function
70
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If $f_n$, f are isometries, then $\underset{n\to\infty}{\lim}\overset{\infty}{\underset{i=0}{\sum}}2^{-i-1}d(f_n(x_i), f(x_i))=0$.

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#sequences-and-series #general-topology #metric-spaces #isometry #sequence-of-function
107
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Check that $\sum_{n=0}^{\infty}(1-x^2)x^n$ uniformly converges on $[0;1]$

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#real-analysis #sequences-and-series #solution-verification #uniform-convergence #sequence-of-function
65
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I wonder why the author assumed $(f_n)_{n\in\mathbb{N}}$ converges pointwise to a function $f$ on $[a,b]$. ("Calculus I" by Shizuo Miyazima)

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#calculus #uniform-convergence #sequence-of-function
54
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Can the monotonicity criterion be dropped from the Abel's Test for Uniform convergence?

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#real-analysis #uniform-convergence #sequence-of-function
38
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Given $X = (C[0,1], d_\infty)$ and $f_n \in X$ then for every $K \subseteq X$ compact, $f_n \nsubseteq K$

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#real-analysis #compactness #sequence-of-function
41
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Can a convergent sequence have divergent subsequences?

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#sequences-and-series #convergence-divergence #sequence-of-function
37
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Differentiability of the limits of sequences of functions from $\mathbb R$ to a Banach space

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#functional-analysis #derivatives #solution-verification #banach-spaces #sequence-of-function
58
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Differentiability of the limits of sequences of functions between Banach spaces

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#functional-analysis #solution-verification #banach-spaces #sequence-of-function #frechet-derivative
58
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Pointwise convergence of a function where convergence to multiple values occur at a single point

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#sequences-and-series #uniform-convergence #sequence-of-function #pointwise-convergence
78
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Rudin $7.14$ linked to $7.9$ why does Rudin make boundedness compulsory in uniform convergence?

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#metric-spaces #uniform-convergence #sequence-of-function
31
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$f_n$ is continuous on $[a,b]$, and for any $x\in [a,b]$, $\{f_n(x)\}$ is a bounded sequence. Show that for some interval, $f_n$ is uniformly bounded

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#calculus #sequences-and-series #sequence-of-function

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