$f in C_{00}(mathbb{R^p},mathbb{C})$. $ mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$...
$begingroup$
Continuing from here
Let $f_t(x):=f(x+t)$
Consider $f mapsto f_t$ which is a linear, isometric bijection from $L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})to L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ for every $t in mathbb{R}^p$
How can I show that for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) the mapping $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous?
real-analysis analysis lp-spaces
asked Jan 5 at 13:56
user626880user626880
93
$endgroup$
add a comment |
$begingroup$
Continuing from here
Let $f_t(x):=f(x+t)$
Consider $f mapsto f_t$ which is a linear, isometric bijection from $L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})to L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ for every $t in mathbb{R}^p$
How can I show that for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) the mapping $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous?
real-analysis analysis lp-spaces
asked Jan 5 at 13:56
user626880user626880
93
$endgroup$
add a comment |
$begingroup$
Continuing from here
Let $f_t(x):=f(x+t)$
Consider $f mapsto f_t$ which is a linear, isometric bijection from $L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})to L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ for every $t in mathbb{R}^p$
How can I show that for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) the mapping $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous?
real-analysis analysis lp-spaces
asked Jan 5 at 13:56
user626880user626880
93
$endgroup$
Continuing from here
Let $f_t(x):=f(x+t)$
Consider $f mapsto f_t$ which is a linear, isometric bijection from $L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})to L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ for every $t in mathbb{R}^p$
How can I show that for $f in C_{00}(mathbb{R^p},mathbb{C})$ (continuous and compact support) the mapping $mathbb{R}^pni t mapsto f_t in L_infty(mathbb{R}^p, mathcal B_p, lambda_p, mathbb{C})$ is uniformly continuous?
real-analysis analysis lp-spaces
real-analysis analysis lp-spaces
asked Jan 5 at 13:56
user626880user626880
93
asked Jan 5 at 13:56
user626880user626880
93
asked Jan 5 at 13:56
user626880user626880
93
asked Jan 5 at 13:56
user626880user626880
93
asked Jan 5 at 13:56
user626880user626880
93
93
add a comment |
add a comment |
1 Answer
1
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$begingroup$
This follows from the fact that a continuous function with compact support is uniformly continuous. For a fixed $varepsilon$, there exists $delta$ such that if $s,tinmathbb R^p$ satisfy $lVert t-srVertltdelta$, then $leftlvert f(t)-f(s)rightrvertltvarepsilon$.
For any $xinmathbb R^p$ and $s,t$ satisfying $lVert t-srVertltdelta$, the following inequality holds
$$
leftlvert f(x+t)-f(x+s)rightrvertltvarepsilon
$$
hence
$$leftlVert f_t-f_srightrVert_infty ... $$
answered Jan 5 at 14:07
Davide GiraudoDavide Giraudo
125k16150261
$endgroup$
add a comment |
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$begingroup$
This follows from the fact that a continuous function with compact support is uniformly continuous. For a fixed $varepsilon$, there exists $delta$ such that if $s,tinmathbb R^p$ satisfy $lVert t-srVertltdelta$, then $leftlvert f(t)-f(s)rightrvertltvarepsilon$.
For any $xinmathbb R^p$ and $s,t$ satisfying $lVert t-srVertltdelta$, the following inequality holds
$$
leftlvert f(x+t)-f(x+s)rightrvertltvarepsilon
$$
hence
$$leftlVert f_t-f_srightrVert_infty ... $$
answered Jan 5 at 14:07
Davide GiraudoDavide Giraudo
125k16150261
$endgroup$
add a comment |
$begingroup$
This follows from the fact that a continuous function with compact support is uniformly continuous. For a fixed $varepsilon$, there exists $delta$ such that if $s,tinmathbb R^p$ satisfy $lVert t-srVertltdelta$, then $leftlvert f(t)-f(s)rightrvertltvarepsilon$.
For any $xinmathbb R^p$ and $s,t$ satisfying $lVert t-srVertltdelta$, the following inequality holds
$$
leftlvert f(x+t)-f(x+s)rightrvertltvarepsilon
$$
hence
$$leftlVert f_t-f_srightrVert_infty ... $$
answered Jan 5 at 14:07
Davide GiraudoDavide Giraudo
125k16150261
$endgroup$
add a comment |
$begingroup$
This follows from the fact that a continuous function with compact support is uniformly continuous. For a fixed $varepsilon$, there exists $delta$ such that if $s,tinmathbb R^p$ satisfy $lVert t-srVertltdelta$, then $leftlvert f(t)-f(s)rightrvertltvarepsilon$.
For any $xinmathbb R^p$ and $s,t$ satisfying $lVert t-srVertltdelta$, the following inequality holds
$$
leftlvert f(x+t)-f(x+s)rightrvertltvarepsilon
$$
hence
$$leftlVert f_t-f_srightrVert_infty ... $$
answered Jan 5 at 14:07
Davide GiraudoDavide Giraudo
125k16150261
$endgroup$
This follows from the fact that a continuous function with compact support is uniformly continuous. For a fixed $varepsilon$, there exists $delta$ such that if $s,tinmathbb R^p$ satisfy $lVert t-srVertltdelta$, then $leftlvert f(t)-f(s)rightrvertltvarepsilon$.
For any $xinmathbb R^p$ and $s,t$ satisfying $lVert t-srVertltdelta$, the following inequality holds
$$
leftlvert f(x+t)-f(x+s)rightrvertltvarepsilon
$$
hence
$$leftlVert f_t-f_srightrVert_infty ... $$
answered Jan 5 at 14:07
Davide GiraudoDavide Giraudo
125k16150261
answered Jan 5 at 14:07
Davide GiraudoDavide Giraudo
125k16150261
answered Jan 5 at 14:07
Davide GiraudoDavide Giraudo
125k16150261
answered Jan 5 at 14:07
Davide GiraudoDavide Giraudo
125k16150261
125k16150261
add a comment |
add a comment |
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