If $e_1, e_2, e_3, ..., e_n$,are the Schauder basis of the normed space (E, ||.||) with the property that $||en|| = 1$ for all $n ∈ N$, and $(v_i)i∈N ⊂ E$ is a sequence such that $lim_{i→∞} ||v_i|| = 0$, must the vector vi coordinates with respect to the Schauder basis $e_1, e_2, e_3, ..., e_n$,necessarily approach zero as i increases to infinity?
My attempt:
The Schauder basis is a sequence of vectors in a normed space that has the property that every vector in the space can be represented as an infinite linear combination of the basis vectors. In this case, the norm of each basis vector is equal to 1.
Now, let's consider the sequence $(v_i)i∈N$ where $lim_{i→∞} ||v_i|| = 0$. This means that as i approaches infinity, the norm of $v_i$ goes to zero. We want to determine if the coordinates of $v_i$ with respect to the Schauder basis also approach zero.
The coordinates of a vector with respect to a basis are obtained by expressing the vector as a linear combination of the basis vectors. If we denote the coordinates of vi with respect to the Schauder basis as $(a_i)_n$, where $a_i$ is the coefficient of the nth basis vector, then we can write: $v_i=\sum_{n=1}^{\infty}a_{i,n}*e_n$
Now, let's analyze whether the coordinates $(a_i)_n$ approach zero as i goes to infinity.
Since $lim_{i→∞} ||v_i|| = 0$, we have:
$lim_{i→∞}||\sum_{n=1}^{\infty}a_{i,n}*e_n||=0$
Now, consider the individual terms in the sum. Since $||e_n|| = 1$ for all n, we have:
$lim_{i→∞}||a_{i,n}*e_n||=lim_{i→∞}|a_{i,n}|*||e_n||=lim_{i→∞}|a_{i,n}|$
Since the limit of $||v_i||$ is zero, it implies that the limit of the individual terms in the sum must be zero. Therefore, we have:
$lim_{i→∞}|a_{i,n}|=0$
This implies that the coordinates $(a_i)_n$ must approach zero as i goes to infinity for each fixed n. In other words, the vector $v_i$, when expressed in terms of the Schauder basis, has coordinates that approach zero as i goes to infinity.
So, the answer is yes, the vector vi coordinates with respect to the Schauder basis must necessarily approach zero as i increases to infinity.
Is this reasoning correct?