Let $X$ be a normed linear space and $X_0$ be a finite dimensional subspace of $X$. Prove that there exists a projection operator $P \in B(X)$ such that $R(P ) = X_0$.
My Try: $X$ can always be written as $X_1 + X_0$ where $X_1$ is a closed subspace and $X_0 \cap X_1 = \{0\}$. So $x \in X, x = y+z, y \in X_0, z\in X_1.$ Define the map $P(x)=y$. I think that this map works.
Can someone check the argument is correct or not? And how to show $P \in B(X)$?
Best Answer
Let $X_0 \le X$ be a finite-dimensional subspace of $X$ and let $\{e_1, \ldots, e_n\}$ be a basis for $X_0$. Then every $x \in X_0$ has a unique representation as a linear combination of $e_1, \ldots, e_n$:
$$x = \alpha_1(x)e_1 + \ldots \alpha_n(x)e_n$$
Notice that $\alpha_i : X_0 \to \mathbb{F}$ are linear functionals, and they are bounded because since functionals on a finite-dimensional space are always bounded.
Thus, using the Hahn-Banach theorem, we can extend them to bounded functionals $\tilde\alpha_i : X \to \mathbb{F}$.
Now define $P : X \to X_0$ as
$$Px = \tilde\alpha_1(x)e_1 + \ldots \tilde\alpha_n(x)e_n, \quad x\in X$$
Notice that $P$ acts as identity on $X_0$:
$$Px = \tilde\alpha_1(\underbrace{x}_{\in X_0})e_1 + \ldots \tilde\alpha_n(\underbrace{x}_{\in X_0})e_n = \alpha_1(x)e_1 + \ldots \alpha_n(x)e_n = x, \quad x \in X_0$$
Therefore, $\DeclareMathOperator{\Ima}{Im}$$P^2 = P$ since $\Ima P = X_0$.
Finally, $P$ is bounded:
\begin{align} \|Px\| &= \|\tilde\alpha_1(x)e_1 + \ldots \tilde\alpha_n(x)e_n\| \\ &\le \|\tilde\alpha_1(x)\|\|e_1\| + \ldots \|\tilde\alpha_n(x)\|\|e_n\| \\ &\le \|\tilde\alpha_1\|\|x\|\|e_1\| + \ldots \|\tilde\alpha_n\|\|x\|\|e_n\| \\ &= \big(\|\tilde\alpha_1\|\|e_1\| + \ldots \|\tilde\alpha_n\|\|e_n\|\big)\|x\| \end{align}
Thus, $P$ is the desired projection.
From $\DeclareMathOperator{\Ker}{Ker}$here we also get the decomposition $X = X_0 \dot+ \Ker P = X_0 \dot+ \bigcap_{i=1}^n \Ker\tilde\alpha_i$. Thus follows that a finite-dimensional subspace always has a direct complement, a fact which cannot be used before conducting a proof similar to this one.