Intereting Posts

Why Doesn't This Series Converge?
Stone-Čech compactification. A completely regular topological space is locally compact iff it is open in its Stone-Čech compactification.
van der pol equation
Gödel's Completeness Theorem and logical consequence
Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis
How do the closed subsets in the product topology look like
If $p\in R$ is irreducible, is it still irreducible in $R$?
Prove that $512^3 + 675^3 + 720^3$ is a composite number
How can I find the number of the shortest paths between two points on a 2D lattice grid?
Ring of holomorphic functions
What's the proof that the Euler totient function is multiplicative?
$A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
$\int_{-\infty}^\infty e^{ikx}dx$ equals what?
Why not to extend the set of natural numbers to make it closed under division by zero?
$f(x-y)$ considered as a function of $(x,y)\in \mathbb{R^{2n}}$ is measurable if $f$ is measurable

Suppose that we have a vector space $X$ over the field $\mathbb F \in \{ \mathbb R, \mathbb C \}$.

**Question:** Does there exist a Hilbert space $\widehat X$ over $\mathbb F$ such that $\widehat X$, when we forget the Hilbert space structure, is the same as $X$ in the category of vector spaces?

Which criteria hold for the existence of such a Hilbert space, on which choices do we have? My motivation is to better understand which structure we gain from the Hilbert space setting (and topological vector spaces in general).

- About the $p$ summable sequences
- Gradient Estimate - Question about Inequality vs. Equality sign in one part
- Why is such an operator continuous?
- Show $T$ is invertible if $T'$ is invertible where $T\in B(X)$, $T'\in B(X')$
- Understanding Arzelà–Ascoli theorem
- Function invariant under Hilbert transform

- Affine transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?
- Graph of symmetric linear map is closed
- linear dependence proof using subsets
- Compute the derivative of the log of the determinant of A with respect to A
- Determinant of the transpose via exterior products
- Anybody knows a proof of Uniqueness of the Reduced Echelon Form Theorem?
- Proving the inverse of a matrix equals $I_n-\frac{1}{n-1}A$
- What's special about the first vector
- Why does Friedberg say that the role of the determinant is less central than in former times?
- Prove the following summation inequality

Let’s go ahead and prove Eric Wofsey’s claim (see his comment under the question), that if $H$ is an infinite-dimensional Hilbert space with an orthonormal basis $U$ of cardinality $\kappa$, then any Hamel basis $B$ of $H$ has cardinality $|B| = \kappa^{\aleph_0}$. Notice this implies that some (real or complex) vector spaces do not admit a Hilbert space structure; e.g., any whose dimension is the limit $\lambda$ of cardinals $\alpha_n$ defined by $\alpha_0 = \aleph_0$ and $\alpha_{n+1} = 2^{\alpha_n}$ (this $\lambda$ has countable cofinality and hence can’t be of the form $\lambda = \kappa^{\aleph_0}$, else $\lambda = \lambda^{\aleph_0}$ which would contradict König’s theorem, as mentioned in an earlier comment).

First remark: $|B| = |H|$. Since $H$ is infinite-dimensional, we have $|B| \geq 2^{\aleph_0} = |\mathbb{C}|$ (see for example the last two paragraphs of this M.SE post: https://math.stackexchange.com/a/547888/43208), and we have $|H| = |B|\cdot |\mathbb{C}|$ by the lemma at this MO post: https://mathoverflow.net/a/49572/2926). Combining these two, $|B| = |B| \cdot |B| \geq |B| \cdot |\mathbb{C}| = |H|$, and obviously $|B| \leq |H|$, so $|B| = |H|$.

Let $\langle -, -\rangle$ be the inner product. For each $h \in H$, let $N_h = \{e \in U: \langle h, e\rangle \neq 0\}$; notice this is at most countable. Consider $H_{fin} = \{h \in H: |N_h| < \infty\}$ and its complement, which I will denote as $H_\infty$.

Each $h \in H_\infty$ determines and is uniquely determined by the countably infinite set $N_h \subseteq U$ together with the corresponding element $\sum_{e \in N_h} \langle h, e\rangle e$ in $l^2(N_h)$. In this way we have a bijection

$$H_\infty \to \bigcup_{N \in X} L(N) \qquad (1)$$

where $X$ is the collection of countably infinite subsets $N$ of $U$ and $L(N)$ is the subset of $l^2(N)$ whose elements $h$ satisfy $\langle h, e\rangle \neq 0$ for all $e \in N$.

**Lemma:** If $A$ is an infinite set and $\alpha$ a cardinal such that $\alpha \leq |A|$, then the number of subsets of $A$ of size $\alpha$ is $|A|^\alpha$.

**Proof:** For each inclusion of a subset of size $\alpha$, we can choose an injective function $\alpha \to A$ with the same image, belonging to the set of all such functions which has cardinality $|A|^\alpha$, so $|A|^\alpha$ is an upper bound. On the other hand, a function $\alpha \to A$ can be identified with its graph, a subset of $\alpha \times A$ of size $\alpha$, and $|\alpha \times A| = |A|$ (excepting $\alpha = 0$ which is trivial), so $|A|^\alpha$ is also a lower bound. $\Box$

Applying this lemma to the bijection (1), we have $|X| = \kappa^{\aleph_0}$. Also we have $|L(N)| = 2^{\aleph_0}$ for each $N$, so $|H_\infty| = \kappa^{\aleph_0} \cdot 2^{\aleph_0} = \kappa^{\aleph_0}$.

Similarly, we have an injection

$$H_{fin} \to \bigcup_{n \geq 0} X_n \times \mathbb{C}^n \qquad (2)$$

where $X_n$ is the collection of subsets of $U$ of finite cardinality $n$. Since $|X_n| = \kappa^n = \kappa$ by the lemma, we have $|H_{fin}| \leq \aleph_0 \cdot \kappa \cdot 2^{\aleph_0} = \max\{\kappa, 2^{\aleph_0}\} \leq \kappa^{\aleph_0}$.

Finally, $\kappa^{\aleph_0} = |H_\infty| \leq |H| = |H_\infty| + |H_{fin}| \leq \kappa^{\aleph_0} + \kappa^{\aleph_0} = \kappa^{\aleph_0}$, so $|H| = \kappa^{\aleph_0}$ as claimed.

**Added in response to Sushil’s comment below:** The converse, that if $\kappa = \kappa^{\aleph_0}$ then there exists a Hilbert space of algebraic dimension $\kappa$, is fairly immediate. Indeed, suppose $B$ is a set of cardinality $\kappa$, and let $V$ be the $\mathbb{C}$-vector space $V$ consisting of formal $\mathbb{C}$-linear combinations of elements of $B$. Assign to $V$ the unique inner product that makes $B$ an orthonormal set of $V$. Then the completion of $V$ with respect to the norm of the inner product is a Hilbert space $H$ whose algebraic dimension is $\kappa$. This is because $B$ is an orthonormal basis of $H$ (by construction of $H$), so the claim of the very first paragraph of this answer shows that any Hamel basis of $H$ has cardinality $\kappa^{\aleph_0} = \kappa$.

Arthur Kruse showed in “Badly incomplete normed linear spaces” *(Math. Z. 83 (1964) 314–320, DOI:10.1007/BF01111164, also freely available at GDZ)* that a Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$.

(This has been reproved several times since then, and perhaps there are even older references.)

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- A trigonometic integral with complex technicals
- Proving $-\frac{1}{2}(z+\frac{1}{z})$ maps upper half disk onto upper half plane
- Prove that $\displaystyle \lim_{x \to \infty} f'(x) = 0$
- Chance on winning by throwing a head on first toss.
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