Intereting Posts

$G_\delta$ sets
Fundamental theorem of Morse theory for $\Omega(S^n )$
Proof of “every convex function is continuous”
Show that every $n$ can be written uniquely in the form $n = ab$, with $a$ square-free and $b$ a perfect square
Relation between quadratic inverse and it's roots
Intersection between conic and line in homogeneous space
Show non-definability of (biparted graphs, graphs with eulerian cycle, even number of nodes)
Showing that a function is of class $C^{\infty}$
G is group of order pq, pq are primes
If a and b are non-negative real numbers then demonstrate inequality
Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$.
Why is this proposition provable and its converse not?
An expression that vanishes over every field
Right-adjoint to the inverse image functor
cardinality of all real sequences

Essentially what the title says – where to me a Hilbert space is a complete (Hermitian) inner product space, am I safe to assume every such real Hilbert space is of uncountable dimension over $\mathbb{R}$, or is there a countable-dimension example?

Thanks a lot ðŸ™‚

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- Find two constant$c_1, c_2$ to make the inequality hold true for all N
- Show that the LU decomposition of matrices of the form $\begin{bmatrix}0& x\\0 & y\end{bmatrix}$ is not unique

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- Show that $f(x) = 0$ for all $x \in $ given $|f'(x)| \leq C|f(x)| $

An infinite dimensional (real) Hilbert space has dimension at least $\mathfrak{c}=2^{\aleph_0}=|\mathbb{R}|$ as a vector space. One way to see this is by taking an orthornormal sequence $e_1,e_2,\ldots$, and considering the linearly independent set $\{\sum_{k=1}^\infty t^ke_k:0\lt t\lt 1\}$.

The same fact extends to Banach spaces, but there orthogonality cannot be used to write so short of a proof. A proof is given in this short article by Lacey.

To just see that the dimension cannot be countable, you could use Baire’s theorem.

For more on this in the Hilbert space case, see Problem 7 of Halmos’s *Hilbert space problem book*. (It is assumed there that the Hilbert spaces are complex instead of real, but this does not affect your question.)

I think natural example of Hilbert space with a countable basis is a space of $L^2([-\pi,\pi])$ with a natural fourier set of basis functions ${e^{inx}, i \in -\infty, \ldots, \infty}$. See https://en.wikipedia.org/wiki/Fourier_series

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