----- > [!proposition] Proposition. ([[every infinite-dimensional normed vector space has a discontinuous linear functional]]) > Every [[dimension|infinite-dimensional]] [[norm|normed]] [[vector space]] has a [[continuous|discontinuous]] [[dual vector space|linear functional]]. ^proposition > [!proof]- Proof. ([[every infinite-dimensional normed vector space has a discontinuous linear functional]]) > Suppose $V$ is an infinite-dimensional normed vector space. [[Every vector space has a basis]] (as follows from [[Zorn's lemma]]); fix a [[basis]] $\{ e_{k} \}_{k \in \Gamma}$ for $V$. Because $V$ is infinite-dimensional, $\Gamma$ is not a finite set. Thus we can assume $\mathbb{N} \subset \Gamma$ (by relabeling a [[countably infinite|countable subset]] of $\Gamma$). Define a [[dual vector space|linear functional]] $\varphi:V \to \mathbb{F}$ by setting $\varphi(e_{j}):=j \|e_{j}\|$ for $j \in \mathbb{N}$, setting $\varphi(e_{j})=0$ for $j \in \Gamma-\mathbb{N}$, and [[linear map|extending linearly]]. Because $\varphi(e_{j}) =j \|e_{j}\|$ for each $j \in \mathbb{N}$, $\varphi$ is [[operator norm|unbounded]], [[characterizing continuity of linear maps|completing the proof]]. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```