Let $L$ be a (topological) complement of $G$. Is $\dim G = \operatorname{codim} L$?

Question

Two Hamel bases of the same vector space have the same cardinality, so we define the dimension of a vector space as the cardinality of one of its Hamel basis.

Could you confirm if my below understanding is fine?

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Let $E$ be an infinite-dimensional Banach space. Let $G$ be a closed subspace of $E$. Let $L$ be a (topological) complement of $G$, i.e., $L$ is a closed subspace of $E$ such that $G\cap L = \{0\}$ and $G+L=E$.

• By cardinal arithmetic, $\dim G +\dim L = \dim E$. This identity implies a result (for Hilbert spaces) in this question.
• By rank-nullity theorem, $\operatorname{codim} L + \dim L = \dim E$.

Then $\dim G = \operatorname{codim} L$ follows from below Lemma.

Lemma Let $A, B, C$ be non-empty sets. If $|A|+|B| = |A|+|C|$, then $|B|=|C|$. Here $|A|$ is the cardinality of $A$.

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As indicated in a comment, above Lemma is not true. I would like to ask if $\dim G = \operatorname{codim} L$ still holds.

Answer 1

Thanks to this enlightening comment by Jochen, I have come up with below proof.

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We need some results from Brezis' Functional Analysis, i.e.,

Theorem 2.10. Let $E$ be a Banach space. Assume that $G$ and $L$ are two closed linear subspaces such that $G+L$ is closed. Then there exists a constant $C \geq 0$ such that $$ \left\{\begin{array}{l} \text { every } z \in G+L \text { admits a decomposition of the form } \\ z=x+y \text { with } x \in G, y \in L,\|x\| \leq C\|z\| \text { and }\|y\| \leq C\|z\| \end{array}\right. $$

Definition. Let $G \subset E$ be a closed subspace of a Banach space $E$. A subspace $L \subset E$ is said to be a topological complement or simply a complement of $G$ if (i) $L$ is closed, (ii) $G \cap L=\{0\}$ and $G+L=E$.

We shall also say that $G$ and $L$ are complementary subspaces of $E$. If this holds, then every $z \in E$ may be uniquely written as $z=x+y$ with $x \in G$ and $y \in L$. It follows from Theorem 2.10 that the projection operators $z \mapsto x$ and $z \mapsto y$ are continuous linear operators. (That property could also serve as a definition of complementary subspaces.)

Let $\pi:E \to G$ be the continuous linear projection map defined in above paragraph. Then $\pi$ is surjective homomorphism such that $\ker \pi = L$. By first isomorphism theorem for groups, $$ \operatorname{im} \pi \cong E / \ker \pi. $$

It follows that $G \cong E/L$ and thus $\dim G = \dim (E/L) =: \operatorname{codim} L$.