Chapter 5 (Folland Real Analysis )Problem 52. (parts (a) and (b) )
I am really clueless about the first 2 parts of problem this problem as to how to approach it. I guess that we do not have to apply the results of section 5.4.
Let $\mathcal{X}$ be a Banach space and $f_{1},\ldots,f_{n}$ be linearly independent elements of $\mathcal{X}^{\star}$. Now
a) Define $T:\mathcal{X}\to \mathbb{C}^{n}$ by $Tx = (f_{1}(x),\ldots,f_{n}(x))$. If $N = \{x: Tx = 0\}$ and $\mathcal{M}$ is the linear span of $f_{1},\ldots,f_{n}$, then $\mathcal{M} = \mathcal{N}^{0}$, where $\mathcal{N}^{0} = \{f\in\mathcal{X}^{\star}: f|\mathcal{N} = 0\}$. Hence $\mathcal{M}^{\star}$ is isomorphic to $(\mathcal{X}/\mathcal{N})^{\star}$.
b) If $F\in \mathcal{X}^{\star\star}$, for any $\epsilon >0 $ there exists $x\in \mathcal{X}$ such that $F(f_{j}) = f_{j}(x)$ for $j = 1,\ldots, n$ and $||x ||\leq (1+\epsilon)||F||$
($F|\mathcal{M}$ can be identified with an element of $(\mathcal{X}/\mathcal{N})^{\star\star}$ and hence an element of $\mathcal{X}/\mathcal{N}$, since the latter is finite dimensional.
for part (b), I did not get the last part, i.e. how does definition of ||x+N|| to obtain the inequality ||x+N||≥||x||(1+ϵ) . Note that since F is fixed (i.e. x is fixed, the above inequality should be true for all x and not for some x.)