MathDB

Functional value at x is taken at points converging to x

Source: Miklós Schweitzer 2017, problem 9

January 13, 2018

normed vector spaceFunctional Analysisanalysiscollege contestsadvanced fields

Problem Statement

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)

M

converging to

x

satisfying

L_j(y_n)=L_j(x)

for all

j=1,\ldots,m

and

n\in \mathbb{N}