FA27

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Let X\, be a normed space and \{x_{1},...,x_{n}\}\, be a linearly independent subset of X\,. Use the Hahn-Banach extension theorem to show that there exist bounded linear functionals f_{1},...,f_{n}\, on X\, such that

f_{i}(x_{j})=\delta _{{i,j}}\,

where \delta _{{i,j}}=0\, when i\neq j\, and 1\, when i=j\,. Can you think about extending this to an infinite sequence \left\{x_{1},...,x_{n},...\right\}\, of vectors?