A question about closure and linear transformation
$begingroup$
I use the following definition of closure:
begin{equation}
text{cl}~C = bigcap_{varepsilon>0}(C + varepsilon B),
end{equation}
where $B$ is Euclidean unit ball: $B = {boldsymbol{x} mid |boldsymbol{x}| leq 1}$.
Suppose that $C$ is subset of $boldsymbol{R}^n$, and linear transformation $mathcal{A}: boldsymbol{R}^n rightarrow boldsymbol{R}^m$.
Hence, is the following formula true?
begin{equation}
mathcal{A}(text{cl}~C) = bigcap_{varepsilon}(mathcal{A}C + varepsilon mathcal{A}B).
end{equation}
I know that $mathcal{A}(C_1 cap C_2) neq mathcal{A}C_1 cap mathcal{A}C_2$ (Intersection of linear Transformation.).
Therefore in this special situation, could i prove it or provide an counter example? Thanks.
real-analysis convex-analysis
asked Dec 18 '18 at 5:15
Ze-Nan LiZe-Nan Li
286
$endgroup$
add a comment |
$begingroup$
I use the following definition of closure:
begin{equation}
text{cl}~C = bigcap_{varepsilon>0}(C + varepsilon B),
end{equation}
where $B$ is Euclidean unit ball: $B = {boldsymbol{x} mid |boldsymbol{x}| leq 1}$.
Suppose that $C$ is subset of $boldsymbol{R}^n$, and linear transformation $mathcal{A}: boldsymbol{R}^n rightarrow boldsymbol{R}^m$.
Hence, is the following formula true?
begin{equation}
mathcal{A}(text{cl}~C) = bigcap_{varepsilon}(mathcal{A}C + varepsilon mathcal{A}B).
end{equation}
I know that $mathcal{A}(C_1 cap C_2) neq mathcal{A}C_1 cap mathcal{A}C_2$ (Intersection of linear Transformation.).
Therefore in this special situation, could i prove it or provide an counter example? Thanks.
real-analysis convex-analysis
asked Dec 18 '18 at 5:15
Ze-Nan LiZe-Nan Li
286
$endgroup$
add a comment |
$begingroup$
I use the following definition of closure:
begin{equation}
text{cl}~C = bigcap_{varepsilon>0}(C + varepsilon B),
end{equation}
where $B$ is Euclidean unit ball: $B = {boldsymbol{x} mid |boldsymbol{x}| leq 1}$.
Suppose that $C$ is subset of $boldsymbol{R}^n$, and linear transformation $mathcal{A}: boldsymbol{R}^n rightarrow boldsymbol{R}^m$.
Hence, is the following formula true?
begin{equation}
mathcal{A}(text{cl}~C) = bigcap_{varepsilon}(mathcal{A}C + varepsilon mathcal{A}B).
end{equation}
I know that $mathcal{A}(C_1 cap C_2) neq mathcal{A}C_1 cap mathcal{A}C_2$ (Intersection of linear Transformation.).
Therefore in this special situation, could i prove it or provide an counter example? Thanks.
real-analysis convex-analysis
asked Dec 18 '18 at 5:15
Ze-Nan LiZe-Nan Li
286
$endgroup$
I use the following definition of closure:
begin{equation}
text{cl}~C = bigcap_{varepsilon>0}(C + varepsilon B),
end{equation}
where $B$ is Euclidean unit ball: $B = {boldsymbol{x} mid |boldsymbol{x}| leq 1}$.
Suppose that $C$ is subset of $boldsymbol{R}^n$, and linear transformation $mathcal{A}: boldsymbol{R}^n rightarrow boldsymbol{R}^m$.
Hence, is the following formula true?
begin{equation}
mathcal{A}(text{cl}~C) = bigcap_{varepsilon}(mathcal{A}C + varepsilon mathcal{A}B).
end{equation}
I know that $mathcal{A}(C_1 cap C_2) neq mathcal{A}C_1 cap mathcal{A}C_2$ (Intersection of linear Transformation.).
Therefore in this special situation, could i prove it or provide an counter example? Thanks.
real-analysis convex-analysis
real-analysis convex-analysis
asked Dec 18 '18 at 5:15
Ze-Nan LiZe-Nan Li
286
asked Dec 18 '18 at 5:15
Ze-Nan LiZe-Nan Li
286
edited Dec 18 '18 at 10:29
Ze-Nan Li
asked Dec 18 '18 at 5:15
Ze-Nan LiZe-Nan Li
286
asked Dec 18 '18 at 5:15
Ze-Nan LiZe-Nan Li
286
asked Dec 18 '18 at 5:15
Ze-Nan LiZe-Nan Li
286
286
add a comment |
add a comment |
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