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Tomer
Tomer
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Let X be an infinite dimensional normed linear space. A sequence $(e_n)$ in X$X$ is called a basis if for every $x \in X$ there is a unique sequence of scalars $(a_n)$ such that $x=\sum_n a_n e_n $ (equality in norm). We say that $(e_n)$ is a Weak basis if for every $x \in X$ there is a unique sequence of scalars $(a_n)$ such that $\sum_{k=1}^n a_k e_k $ converges weakly to x as $n \to \infty$.

I am looking for a reference to the following theorem, which i have seen in several places, yet was not able to find a proof, nor to come up with one myself: if $(e_n)$ is a weak basis, then it is a basis.

Let X be an infinite dimensional normed linear space. A sequence $(e_n)$ in X is called a basis if for every $x \in X$ there is a unique sequence of scalars $(a_n)$ such that $x=\sum_n a_n e_n $ (equality in norm). We say that $(e_n)$ is a Weak basis if for every $x \in X$ there is a unique sequence of scalars $(a_n)$ such that $\sum_{k=1}^n a_k e_k $ converges weakly to x as $n \to \infty$.

I am looking for a reference to the following theorem, which i have seen in several places, yet was not able to find a proof, nor to come up with one myself: if $(e_n)$ is a weak basis, then it is a basis.

Let X be an infinite dimensional normed linear space. A sequence $(e_n)$ in $X$ is called a basis if for every $x \in X$ there is a unique sequence of scalars $(a_n)$ such that $x=\sum_n a_n e_n $ (equality in norm). We say that $(e_n)$ is a Weak basis if for every $x \in X$ there is a unique sequence of scalars $(a_n)$ such that $\sum_{k=1}^n a_k e_k $ converges weakly to x as $n \to \infty$.

I am looking for a reference to the following theorem, which i have seen in several places, yet was not able to find a proof, nor to come up with one myself: if $(e_n)$ is a weak basis, then it is a basis.

Source Link
Tomer
Tomer
  • 165
  • 6

Weak basis of normed linear space

Let X be an infinite dimensional normed linear space. A sequence $(e_n)$ in X is called a basis if for every $x \in X$ there is a unique sequence of scalars $(a_n)$ such that $x=\sum_n a_n e_n $ (equality in norm). We say that $(e_n)$ is a Weak basis if for every $x \in X$ there is a unique sequence of scalars $(a_n)$ such that $\sum_{k=1}^n a_k e_k $ converges weakly to x as $n \to \infty$.

I am looking for a reference to the following theorem, which i have seen in several places, yet was not able to find a proof, nor to come up with one myself: if $(e_n)$ is a weak basis, then it is a basis.