Uniform convergence: how to write it correctly

Question

Is $f_{n}\underset{n\rightarrow \infty }{\longrightarrow }f$ equivalent to $f_{n}\left( x\right) \underset{n\rightarrow \infty }{% \longrightarrow }f\left( x\right) $ ?

When we have a sequence of functions $\left( f_{n}\right) _{n\in \mathbb{N}}$ that converges uniformly to $f,$ we depict this by $f_{n}\underset{% n\rightarrow \infty }{\longrightarrow }f.$ Is it ok if we also write $% f_{n}\left( x\right) \underset{n\rightarrow \infty }{\longrightarrow }% f\left( x\right) $ ? I have a feeling something is not exactly right.

Answer 1

Notation itself has no meaning until one defines it. There is no "the" right way to write anything. There are conventions to follow though.

One usually describes uniform convergence as

$f_n\to f$ uniformly as $n\to\infty$.

One writes "point-wise" convergence as

For every $x$ (in the domain), $f_n(x)\to f(x)$ as $n\to\infty$.

or simply says

$f_n$ converges to $f$ pointwise everywhere in the domain.

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Is it OK if we also write ...

One rarely uses such notation for uniform convergence.