I am solving assignments of a Institute of which I am not a student as our instructor never gives assignments.
Can somebody please help me with this problem .
Question is -----> Prove that for every normed linear space X,
dimension of X' $\ge$ dimension of X. ( X' is dual space of X)
I am sorry but I am not able to think how to even start this problem. Can somebody please help.
If $\{e_1,\ldots,e_n\}$ is a basis of $X$, every vector $x\in X$ can be decomposed uniquely according to this basis \begin{equation} x = x^1 e_1 +\cdots+x^n e_n \end{equation} The map $f_i: x\mapsto x^i$ that maps every vector to its i-th coordinate on this basis is a linear functional on the space $X$, that is to say $f_i \in X'$.
Hint: Prove that $f_1,\ldots,f_n$ are linearly independent in $X'$.