A functional analysis exam question
up vote
0
down vote
favorite
Let $X$ be the metric space and it is not a compact set.Show that
$(1)$There is $varepsilon>0$ and the sequence $left{ x_n right}subset X$ ,when $mne n$,there is$$Bleft( x_n,varepsilon right) cap Bleft( x_m,varepsilon right)=oslash.$$
$(2)$There is a continuous function $f_n(x):Xlongrightarrow left[ text{0,}1 right]$ for any $n$,such that
$$f_n(x_{n})=1$$if and only if $xnotin Bleft( x,frac{varepsilon}{2} right)$,there is$f_n(x)=0.$
I worked hard but didn't solve it.I started from a definition that is not compact set, but I don't know how to find the sequence $left{ x_n right}$.So I hope you can give me some ideas.
functional-analysis
add a comment |
up vote
0
down vote
favorite
Let $X$ be the metric space and it is not a compact set.Show that
$(1)$There is $varepsilon>0$ and the sequence $left{ x_n right}subset X$ ,when $mne n$,there is$$Bleft( x_n,varepsilon right) cap Bleft( x_m,varepsilon right)=oslash.$$
$(2)$There is a continuous function $f_n(x):Xlongrightarrow left[ text{0,}1 right]$ for any $n$,such that
$$f_n(x_{n})=1$$if and only if $xnotin Bleft( x,frac{varepsilon}{2} right)$,there is$f_n(x)=0.$
I worked hard but didn't solve it.I started from a definition that is not compact set, but I don't know how to find the sequence $left{ x_n right}$.So I hope you can give me some ideas.
functional-analysis
Part (1) is only true if the space $X$ is also assumed to complete.
– s.harp
Nov 19 at 11:09
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $X$ be the metric space and it is not a compact set.Show that
$(1)$There is $varepsilon>0$ and the sequence $left{ x_n right}subset X$ ,when $mne n$,there is$$Bleft( x_n,varepsilon right) cap Bleft( x_m,varepsilon right)=oslash.$$
$(2)$There is a continuous function $f_n(x):Xlongrightarrow left[ text{0,}1 right]$ for any $n$,such that
$$f_n(x_{n})=1$$if and only if $xnotin Bleft( x,frac{varepsilon}{2} right)$,there is$f_n(x)=0.$
I worked hard but didn't solve it.I started from a definition that is not compact set, but I don't know how to find the sequence $left{ x_n right}$.So I hope you can give me some ideas.
functional-analysis
Let $X$ be the metric space and it is not a compact set.Show that
$(1)$There is $varepsilon>0$ and the sequence $left{ x_n right}subset X$ ,when $mne n$,there is$$Bleft( x_n,varepsilon right) cap Bleft( x_m,varepsilon right)=oslash.$$
$(2)$There is a continuous function $f_n(x):Xlongrightarrow left[ text{0,}1 right]$ for any $n$,such that
$$f_n(x_{n})=1$$if and only if $xnotin Bleft( x,frac{varepsilon}{2} right)$,there is$f_n(x)=0.$
I worked hard but didn't solve it.I started from a definition that is not compact set, but I don't know how to find the sequence $left{ x_n right}$.So I hope you can give me some ideas.
functional-analysis
functional-analysis
asked Nov 19 at 10:58
daimengjie
6
6
Part (1) is only true if the space $X$ is also assumed to complete.
– s.harp
Nov 19 at 11:09
add a comment |
Part (1) is only true if the space $X$ is also assumed to complete.
– s.harp
Nov 19 at 11:09
Part (1) is only true if the space $X$ is also assumed to complete.
– s.harp
Nov 19 at 11:09
Part (1) is only true if the space $X$ is also assumed to complete.
– s.harp
Nov 19 at 11:09
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
You need extra hypothesis for $(1)$. For example consider the open interval $(0,1)$. Because the mention to 'functional analysis' in the title I suppose that $X$ is an infinite dimensional normed space or similar.
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
You need extra hypothesis for $(1)$. For example consider the open interval $(0,1)$. Because the mention to 'functional analysis' in the title I suppose that $X$ is an infinite dimensional normed space or similar.
add a comment |
up vote
0
down vote
You need extra hypothesis for $(1)$. For example consider the open interval $(0,1)$. Because the mention to 'functional analysis' in the title I suppose that $X$ is an infinite dimensional normed space or similar.
add a comment |
up vote
0
down vote
up vote
0
down vote
You need extra hypothesis for $(1)$. For example consider the open interval $(0,1)$. Because the mention to 'functional analysis' in the title I suppose that $X$ is an infinite dimensional normed space or similar.
You need extra hypothesis for $(1)$. For example consider the open interval $(0,1)$. Because the mention to 'functional analysis' in the title I suppose that $X$ is an infinite dimensional normed space or similar.
answered Nov 19 at 11:08
Dante Grevino
7367
7367
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004787%2fa-functional-analysis-exam-question%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Part (1) is only true if the space $X$ is also assumed to complete.
– s.harp
Nov 19 at 11:09