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9

Part of 2017 Miklós Schweitzer

Problems(1)

Functional value at x is taken at points converging to x

Source: Miklós Schweitzer 2017, problem 9

1/13/2018
Let NN be a normed linear space with a dense linear subspace MM. Prove that if L1,…,LmL_1,\ldots,L_m are continuous linear functionals on NN, then for all x∈Nx\in N there exists a sequence (yn)(y_n) in MM converging to xx satisfying Lj(yn)=Lj(x)L_j(y_n)=L_j(x) for all j=1,…,mj=1,\ldots,m and n∈Nn\in \mathbb{N}.
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