Relationship between an infinite-dimensional real vector space and its dual space.

Question

Suppose $V$ is an infinite-dimensional vector space. I want to prove the following two claims: (The following $V^*$ representing the algebraic dual)

Prove that there is an injective linear map from $V$ to $V^*$;
Prove that there is no injective linear map from $V^*$ to $V$.

For the first claim, I think it's trivial by the methods we used for the finite-dimensional case, i.e., we define $\{e^i\}$ as the dual basis of the basis $\{e_i\}$. I think this can also be applied to the infinite-dimensional vector space. (am I right?)

However, I don't know how to prove the second claim. Can anyone give me some hints? Maybe we need to calculate the cardinality but I don't know how to use this tool. Thanks for any hints!

Your statement is not true. The statement should be "If $V$ is finite dimensional then $V$ is isomorphic to $V^*$". However, there are infinite dimensional spaces which are isomorphic to their duals, for instance $\ell^2$. — Aniruddha Deshmukh, Oct 28 '21 at 05:30
@AniruddhaDeshmukh You are confusing the algebraic dual with the Banach space dual. The dual space of bounded (continuous) linear operators of $\ell^2$ is isomorphic to $\ell^2$, but the algebraic dual space is of strictly larger Hamel dimension so it is not isomorphic to $\ell^2$. Here's a proof that an infinite dimensional vector space is never isomorphic to its algebraic dual — Arturo Magidin, Oct 28 '21 at 05:35
@ArturoMagidin: You are correct! I was indeed thinking about the normed duals and not the algebraic duals. — Aniruddha Deshmukh, Oct 28 '21 at 05:37

score 2 · Accepted Answer · answered Oct 28 '21 at 08:24

If a real vector space $V$ has a Hamel basis of cardinality $\mathfrak{m}$ then it is isomorphic to the subset of $\mathbb{R}^{\mathfrak{m}}$ consists of al seqences $(x_t)_{t<\mathfrak{m}}$ such that $x_t \neq0 $ only for finite numbers of $t.$ Hence the cardinality of $V$ is equal $\max\{\overline{\overline{\mathbb{R}}},\mathfrak{m}\}.$

Now observe that every sequence $y=(y_t ) \in \mathbb{R}^{\mathfrak{m}}$ defines a linear functional on $V$ in the sense $$y(x) =\sum_{t<\mathfrak{m}} y_t x_t$$ where $x=\sum x_t e_t $ and $\{e_t \}$ is a Hamel basis of $V.$

Thus the cardinality of $V^*$ is greater or equal to $\left(\overline{\overline{\mathbb{R}}}\right)^{\mathfrak{m}}$ which is strictly greater than $\max\{\overline{\overline{\mathbb{R}}},\mathfrak{m}\}=\overline{\overline{\mathbb{V}}}.$ Hence there exists no injective map from $V^* $ to $V.$

Relationship between an infinite-dimensional real vector space and its dual space.

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