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Found problems: 1

2017 Miklós Schweitzer, 9
Let $N$ be a normed linear space with a dense linear subspace $M$. Prove that if $L_1,\ldots,L_m$ are continuous linear functionals on $N$, then for all $x\in N$ there exists a sequence $(y_n)$ in $M$ converging to $x$ satisfying $L_j(y_n)=L_j(x)$ for all $j=1,\ldots,m$ and $n\in \mathbb{N}$.
advanced fields functional analysis analysis normed vector space real analysis sequences