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Since this is too long for a comment, I post it as an answer:

Sorry, but I do not agree with Peter Michor's answer. There are certainly better examples, but this the first I can remember: There are countable inductive limits $X=\lim\limits_{\to} X_n$ of Frechet spaces which are not Hausdorff (take a decreasing sequence of open connected sets $U_n$ in the complex plane with empty intersection and $X_n=H(U_n)$ the Frechet space of holomorphic functions on $U_n$ together with the injective restriction maps). $X$ is the a quotient of the direct sum $\bigoplus X_n$ which is certainly bornological and the kernel of the quotient map is sequentially closed (because convergent sequences in the direct sum are located and convergent in some finite sum) but it is not closed because the quotient is not Hausdorff.

The situation is better for metrizable spaces (of course, this is trivial) as well as for so-called Silva spaces (also called LS or DFS-spaces, countable inductive limits of Banach spaces with compact inclusions): In these cases, sequentially closed subspaces are closed.

Edit. A simpler example (but possibly less relevant for analytical applications) is the space $X=\mathbb R^I$ endowed with the product topology (point-wise convergence of functions $f:I \to \mathbb R$) if $I$ is uncountable and of moderate cardinality (e.g. $I=\mathbb R$). Then $X$ is bornological (due to the cardinality restriction) and $L=\lbrace f\in X: \lbrace i\in I: f(i)\neq 0\rbrace \text{countable}\rbrace$ is sequentially closed and dense in $X$.

Since this is too long for a comment, I post it as an answer:

Sorry, but I do not agree with Peter Michor's answer. There are certainly better examples, but this the first I can remember: There are countable inductive limits $X=\lim\limits_{\to} X_n$ of Frechet spaces which are not Hausdorff (take a decreasing sequence of open connected sets $U_n$ in the complex plane with empty intersection and $X_n=H(U_n)$ the Frechet space of holomorphic functions on $U_n$ together with the injective restriction maps). $X$ is the a quotient of the direct sum $\bigoplus X_n$ which is certainly bornological and the kernel of the quotient map is sequentially closed (because convergent sequences in the direct sum are located and convergent in some finite sum) but it is not closed because the quotient is not Hausdorff.

The situation is better for metrizable spaces (of course, this is trivial) as well as for so-called Silva spaces (also called LS or DFS-spaces, countable inductive limits of Banach spaces with compact inclusions): In these cases, sequentially closed subspaces are closed.

By 8.5.28 in the book of Bonet and Perez-Carreras, Barrelled Locally Convex Spaces, even sequentially closed subsets of Silva spaces are closed.

Edit. A simpler example (but possibly less relevant for analytical applications) is the space $X=\mathbb R^I$ endowed with the product topology (point-wise convergence of functions $f:I \to \mathbb R$) if $I$ is uncountable and of moderate cardinality (e.g. $I=\mathbb R$). Then $X$ is bornological (due to the cardinality restriction) and $L=\lbrace f\in X: \lbrace i\in I: f(i)\neq 0\rbrace \text{countable}\rbrace$ is sequentially closed and dense in $X$.

Since this is too long for a comment, I post it as an answer:

Sorry, but I do not agree with Peter Michor's answer. There are certainly better examples, but this the first I can remember: There are countable inductive limits $X=\lim\limits_{\to} X_n$ of Frechet spaces which are not Hausdorff (take a decreasing sequence of open connected sets $U_n$ in the complex plane with empty intersection and $X_n=H(U_n)$ the Frechet space of holomorphic functions on $U_n$ together with the injective restriction maps). $X$ is the a quotient of the direct sum $\bigoplus X_n$ which is certainly bornological and the kernel of the quotient map is sequentially closed (because convergent sequences in the direct sum are located and convergent in some finite sum) but it is not closed because the quotient is not Hausdorff.

The situation is better for metrizable spaces (of course, this is trivial) as well as for so-called Silva spaces (also called LS or DFS-spaces, countable inductive limits of Banach spaces with compact inclusions): In these cases, sequentially closed subspaces are closed.

Since this is too long for a comment, I post it as an answer:

Sorry, but I do not agree with Peter Michor's answer. There are certainly better examples, but this the first I can remember: There are countable inductive limits $X=\lim\limits_{\to} X_n$ of Frechet spaces which are not Hausdorff (take a decreasing sequence of open connected sets $U_n$ in the complex plane with empty intersection and $X_n=H(U_n)$ the Frechet space of holomorphic functions on $U_n$ together with the injective restriction maps). $X$ is the a quotient of the direct sum $\bigoplus X_n$ which is certainly bornological and the kernel of the quotient map is sequentially closed (because convergent sequences in the direct sum are located and convergent in some finite sum) but it is not closed because the quotient is not Hausdorff.

The situation is better for metrizable spaces (of course, this is trivial) as well as for so-called Silva spaces (also called LS or DFS-spaces, countable inductive limits of Banach spaces with compact inclusions): In these cases, sequentially closed subspaces are closed.

Edit. A simpler example (but possibly less relevant for analytical applications) is the space $X=\mathbb R^I$ endowed with the product topology (point-wise convergence of functions $f:I \to \mathbb R$) if $I$ is uncountable and of moderate cardinality (e.g. $I=\mathbb R$). Then $X$ is bornological (due to the cardinality restriction) and $L=\lbrace f\in X: \lbrace i\in I: f(i)\neq 0\rbrace \text{countable}\rbrace$ is sequentially closed and dense in $X$.

Since this is too long for a comment, I post it as an answer:

Sorry, but I do not agree with Peter Michors answer. There are certainly better examples, but this the first I can remember: There are countable inductive limits $X=\lim\limits_{\to} X_n$ of Frechet spaces which are not Hausdorff (take a decreasing sequence of open connected sets $U_n$ in the complex plane with empty intersection and $X_n=H(U_n)$ the Frechet space of holomorphic functions on $U_n$ together with the injective restriction maps). $X$ is the a quotient of the direct sum $\bigoplus X_n$ which is certainly bornological and the kernel of the quotient map is sequentially closed (because convergent sequences in the direct sum are located and convergent in some finite sum) but it is not closed because the quotient is not Hausdorff.

The situation is better for metrizable spaces (of course, this is trivial) as well as for so-called Silva spaces (also called LS or DFS-spaces, countable inductive limits of Banach spaces with compact inclusions): In these cases, sequentially closed subspaces are closed.

Since this is too long for a comment, I post it as an answer:

Sorry, but I do not agree with Peter Michor's answer. There are certainly better examples, but this the first I can remember: There are countable inductive limits $X=\lim\limits_{\to} X_n$ of Frechet spaces which are not Hausdorff (take a decreasing sequence of open connected sets $U_n$ in the complex plane with empty intersection and $X_n=H(U_n)$ the Frechet space of holomorphic functions on $U_n$ together with the injective restriction maps). $X$ is the a quotient of the direct sum $\bigoplus X_n$ which is certainly bornological and the kernel of the quotient map is sequentially closed (because convergent sequences in the direct sum are located and convergent in some finite sum) but it is not closed because the quotient is not Hausdorff.

The situation is better for metrizable spaces (of course, this is trivial) as well as for so-called Silva spaces (also called LS or DFS-spaces, countable inductive limits of Banach spaces with compact inclusions): In these cases, sequentially closed subspaces are closed.