Let $X$ be a normed space over the field $\mathbb{K}$.
Let $x_0 \in X, x_0 \ne 0$.
- Prove that there exists a closed subspace $M$ of $X$ such that $X = M \oplus \mathbb{K} x_0$.
- Let $X_0$ be a finite dimensional subspace of $X$. Prove there exists a closed subspace $M$ of $X$ such that $X = M \oplus X_0$.
I want to prove the existence of a closed subspace $M$ such that the decomposition $X = M \oplus \mathbb{K} x_0$ holds true. This means that I am looking for a subspace of $X$ of codimension 1, that is the sought subspace $M$ is an hyperplane. As a consequence, $M$ can be written as $M = Ker (f)$ for a certain linear functional $f: X \rightarrow \mathbb{K}$. Furthermore, I know that $M = Ker (f)$ is closed $\Leftrightarrow$ $f \in X'$. If I can exhibit a linear continuous functional $f$, then I can conclude.
Under the hypotheses that $X$ is a normed vector space and that $x_0 \in X \setminus \{0\}$, a corollary to the Hahn-Banach theorem (analytic form) ensures the existence of a functional $f_0 \in X'$ such that: $f_0(x_0) = ||x_0||^2_X$ and $||f||_{X'} = ||x_0||_X$. Thus, $M = Ker (f_0)$ is the sought closed subspace.
In order to demonstrate 2., I proceeded as follows.
Since, by hypothesis, $dim (X_0) < \infty$, every linear functional $f: X_0 \rightarrow \mathbb{K}$ is continuous. Thus, the Hahn-Banach theorem (analytic form) ensures the existence of an extension of $f$, $\tilde{f} \in X'$, which preserves the norm of the functional. However, my reasoning fails here. My idea was to define a subspace $M$ in terms of the action of $\tilde{f}$ on its elements.
[For instance, $M := \{x \in X: \tilde{f}(x) = \alpha\}$ for a given $\alpha \in \mathbb{K}$ is an affine hyperplane since $\tilde{f} \ne 0$ is a linear functional. In addition, $M$ is closed since $\tilde{f}$ is continuous, but $M$ is not a subspace of $X$ (if $x,y \in M$, $f(x+y) = 2\alpha \notin M$).]
I think that the aim of the exercise was to extend the decomposition seen in 1 in a more general case, but I have no clue on how to conclude. How can I adjust the reasoning to find the required subspace $M$?
Any suggestion would be extremely appreciated.
Best Answer
What you can do is the following. Let $x_1,\ldots,x_n$ be a basis of $X_0$. Given $x\in X_0$, there are unique coefficients $f_1(x),\ldots,f_n(x)$ such that $$x=f_1(x)x_1+\cdots+f_n(x)x_n.$$ Now use the uniqueness to show that $f_1,\ldots,f_n$ are linear, and extend by Hahn-Banach.