Is this extension continuous on $X$?
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Let $X$ be a locally convex space. Let $M$ be a dense subspace of $X$ and let $fin M^*$. And I am trying to show there exists $gin X^*$ such that $g|_M=f$.
My attempts are:
Let $xin X$. Then there exists a net $(x_i)$ in $M$ such that $x_irightarrow x$. Thus $(x_i)$ is a Cauchy net in $M$, and hence $(f(x_i))$ is a Cauchy net in $mathbbF$. Since $mathbbF$ is a Banach space, $limlimits_i f(x_i)$ exists in $mathbbF$. Now define $g(x)=limlimits_i f(x_i)$. Then $g:XrightarrowmathbbF$ is linear and $g|_M=f$. Let $(x_k)$ be a net in $X$ such that $x_krightarrow x$.
And I am trying to show $g(x_k)rightarrow g(x)$ to get continuity of $g$. But I am stuck here. Does $g(x_k)rightarrow g(x)$? If so how to prove it?
Thank you in advance!
functional-analysis locally-convex-spaces
asked Sep 11 at 4:13
Answer Lee
54138
add a comment |
up vote
1
down vote
favorite
Let $X$ be a locally convex space. Let $M$ be a dense subspace of $X$ and let $fin M^*$. And I am trying to show there exists $gin X^*$ such that $g|_M=f$.
My attempts are:
Let $xin X$. Then there exists a net $(x_i)$ in $M$ such that $x_irightarrow x$. Thus $(x_i)$ is a Cauchy net in $M$, and hence $(f(x_i))$ is a Cauchy net in $mathbbF$. Since $mathbbF$ is a Banach space, $limlimits_i f(x_i)$ exists in $mathbbF$. Now define $g(x)=limlimits_i f(x_i)$. Then $g:XrightarrowmathbbF$ is linear and $g|_M=f$. Let $(x_k)$ be a net in $X$ such that $x_krightarrow x$.
And I am trying to show $g(x_k)rightarrow g(x)$ to get continuity of $g$. But I am stuck here. Does $g(x_k)rightarrow g(x)$? If so how to prove it?
Thank you in advance!
functional-analysis locally-convex-spaces
asked Sep 11 at 4:13
Answer Lee
54138
Why is $g$ well-defined and linear? I cannot see that.
– amsmath
Sep 11 at 4:30
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $X$ be a locally convex space. Let $M$ be a dense subspace of $X$ and let $fin M^*$. And I am trying to show there exists $gin X^*$ such that $g|_M=f$.
My attempts are:
Let $xin X$. Then there exists a net $(x_i)$ in $M$ such that $x_irightarrow x$. Thus $(x_i)$ is a Cauchy net in $M$, and hence $(f(x_i))$ is a Cauchy net in $mathbbF$. Since $mathbbF$ is a Banach space, $limlimits_i f(x_i)$ exists in $mathbbF$. Now define $g(x)=limlimits_i f(x_i)$. Then $g:XrightarrowmathbbF$ is linear and $g|_M=f$. Let $(x_k)$ be a net in $X$ such that $x_krightarrow x$.
And I am trying to show $g(x_k)rightarrow g(x)$ to get continuity of $g$. But I am stuck here. Does $g(x_k)rightarrow g(x)$? If so how to prove it?
Thank you in advance!
functional-analysis locally-convex-spaces
asked Sep 11 at 4:13
Answer Lee
54138
Let $X$ be a locally convex space. Let $M$ be a dense subspace of $X$ and let $fin M^*$. And I am trying to show there exists $gin X^*$ such that $g|_M=f$.
My attempts are:
Let $xin X$. Then there exists a net $(x_i)$ in $M$ such that $x_irightarrow x$. Thus $(x_i)$ is a Cauchy net in $M$, and hence $(f(x_i))$ is a Cauchy net in $mathbbF$. Since $mathbbF$ is a Banach space, $limlimits_i f(x_i)$ exists in $mathbbF$. Now define $g(x)=limlimits_i f(x_i)$. Then $g:XrightarrowmathbbF$ is linear and $g|_M=f$. Let $(x_k)$ be a net in $X$ such that $x_krightarrow x$.
And I am trying to show $g(x_k)rightarrow g(x)$ to get continuity of $g$. But I am stuck here. Does $g(x_k)rightarrow g(x)$? If so how to prove it?
Thank you in advance!
functional-analysis locally-convex-spaces
functional-analysis locally-convex-spaces
asked Sep 11 at 4:13
Answer Lee
54138
asked Sep 11 at 4:13
Answer Lee
54138
asked Sep 11 at 4:13
Answer Lee
54138
asked Sep 11 at 4:13
Answer Lee
54138
asked Sep 11 at 4:13
Answer Lee
54138
54138
Why is $g$ well-defined and linear? I cannot see that.
– amsmath
Sep 11 at 4:30
add a comment |
Why is $g$ well-defined and linear? I cannot see that.
– amsmath
Sep 11 at 4:30
Why is $g$ well-defined and linear? I cannot see that.
– amsmath
Sep 11 at 4:30
Why is $g$ well-defined and linear? I cannot see that.
– amsmath
Sep 11 at 4:30
add a comment |
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Why is $g$ well-defined and linear? I cannot see that.
– amsmath
Sep 11 at 4:30