Direct-sum of subspaces

Question

A closed subspace $M$ of a normed vector space $X$ is complemented in $X$ if there exists a closed subspace $N$ such that $M \oplus N = X$, i.e., $M + N = X$ and $M \cap N = \{0\}$. Prove that $M$ is complemented in $X$ if $M$ is finite dimensional. Hint: Start by considering a basis of $M^{\ast}$. Prove also that $M$ is complemented in $X$ if $M$ has finite codimension, i.e., $\dim X/M < \infty$.

Answer 1

Let $\{b_1,\cdots,b_n\}$ be a basis of $M$. By Hahn-Banach there is a basis $\{f_1,\cdots,f_n\}$ of $M^*$ such that $f_j(b_k)=\delta_{jk}$.

Now define $P:X \to X$ by $P(x)=\sum\limits_{k=1}^nf_k(x)b_k$.

Show that $P$ is linear and bounded, $P^2=P$ and $P(X)=M$.

Furthermore we have $X=P(X) \oplus Q(X)$, where $Q=I-P$. Now show that $Q(X)=\ker(P)$.

Hence: $X=M \oplus \ker(P)$ and $\ker(P)$ is closed, since $P$ is bounded.