Intereting Posts

Cardinal arithmetic gone wrong?
Defining the Complex numbers
How can I find the surface area of a normal chicken egg?
Prove that $ \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt{abc}}\right)$.
If $\displaystyle \lim _{x\to +\infty}y(x)\in \mathbb R$, then $\lim _{x\to +\infty}y'(x)=0$
What kind of f(n)'s make the limsup statement is true? What kind don't?
Continuity of the derivative at a point given certain hypotheses
Finding all solutions for a system of equations with constraint on the sum of absolute values
Can't understand this pseudo-inverse relation.
Indefinite integral $\int \arcsin \left(k\sin x\right) dx$
Is my solution for proving $3^n > n^2$ using induction correct?
Proof of $f = g \in L^1_{loc}$ if $f$ and $g$ act equally on $C_c^\infty$
Why does an integral change signs when flipping the boundaries?
Prob. 1, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property?
Proof of Pythagorean theorem without using geometry for a high school student?

Let $H_1, H_2, \ldots, H_n$ be a countable family of Hilbert spaces. Let H be the set of tuples $x = (x_1, \ldots, x_n,\ldots)\in \prod_n H_n$

with the property that

$$\|x \| ^2 =\sum_n \| x_n \| _{H_n}^2 <\infty.$$

Then H is also a Hilbert space.

Prove that H is non-separable and determine an orthonormal basis in this space.

If I take a sequence $$(x^{(k)})$$ such that: $$x^{(1)} = (1,0,0,\ldots), x^{(2)} = (0,1,0,\ldots)$$ then this sequence has no converging subsequnce, hence not separable (?). I have no good idea how to find a basis.

- Is $W_0^{1,p}(\Omega)$ complemented in $W^{1,p}(\Omega)$?
- If the dual spaces are isometrically isomorphic are the spaces isomorphic?
- example of compact operator
- Is ${\rm conv}({\rm ext}((C(X))_1))$ dense in $(C(X))_1$?
- An application of J.-L. Lion's Lemma
- Counterexample for the stability of orthogonal projections

- Must every subset of $\mathbb R$ contain $2$ homeomorphic distinct open sets?
- Is every $T_4$ topological space divisible?
- Torus cannot be embedded in $\mathbb R^2$
- Are sequences with Cesaro mean a closed subset of $\ell_\infty$?
- Can spectrum “specify” an operator?
- Every compact space is a continuous image of a compact Moscow space.
- $u\in W^{1,1}(\Omega)$, $f\in C^1(\mathbb{R})$, but: $f\circ u\notin W^{1,1}(\Omega)$
- Orbits of properly discontinuous actions
- The Cantor set is homeomorphic to infinite product of $\{0,1\}$ with itself - cylinder basis - and it topology
- Classifying the compact subsets of $L^p$

I think you are getting confused with the notation here. For each of your $x_n$, its coordinates are vectors in $H_n$; so “$1$” makes no sense there.

Note that we cannot expect the construction to always give a non-separable Hilbert space. Because we can take $H_n=\mathbb C$ for all $n$, and then $\prod_nH_n=\ell^2(\mathbb N)$, which is separable.

Note also that you never defined what the inner product in the direct sum is, but your condition on the norms suggests that it is the canonical one,

$$

\langle x,y\rangle = \sum_n\langle x_n,y_n\rangle.

$$

To make up a basis of $\prod_nH_n$, the natural way is to use bases from each of the $H_n$. So, for each $n$, fix an orthonormal basis $B_n=\{e_{k,n}\}_{k\in K_n}$ of $H_n$.

Let

$$

B=\{\,x\in\prod_nH_n:\ \exists m\text{ with }x_m\in B_m\text{ and }x_r=0\text{ if }r\ne m\}

$$

It is clear that $B$ is an orthonormal set. Now suppose that $x\in B^\perp$. Then, for any $m$ and any $k\in K_m$,

$$

0=\langle x,e_{k,m}\rangle=\langle x_m,e_{k,m}\rangle.

$$

As $k\in K_m$ was arbitrary, $x_m=0$; as $m$ was arbitrary, $x=0$. So $B$ is total, and it is thus a basis.

Now, if all the $H_n$ are separable, then $K_n$ is countable for all $n$. As the family $H_1,H_2,\ldots$ is countable, we have

$$

B=\bigcup_{m\in\mathbb N}\{x:\ x_m\in B_m\text{ and zero elsewhere }\}.

$$

As each $B_m$ is countable, $B$ is a countable union of countable sets, and is itself counatble. So $H$ is separable.

If any of the spaces $H_1,H_2,\ldots$ is non-separable, or if the family $\{H_n\}$ is uncountable, then $\prod H_n$ will be non-separable.

- Why is “glide symmetry” its own type?
- Prove $e^{i \pi} = -1$
- If point is zero-dimensional, how can it form a finite one dimensional line?
- particular solution of $4y''-y= \sin(x)\cdot \cos(x/2)$
- Positive integer solutions to $\frac{(x+y)^{x+y}(y+z)^{y+z}(x+z)^{x+z}}{x^{2x}y^{2y}z^{2z}}=2016^m$
- What is the order of the sum of log x?
- What is the probability that $X<Y$?
- How to solve this integral: $ \int_{-1}^{1} \frac{x^4}{a^x+1}dx $?
- how to solve this PDE
- The inverse of a matrix in which the sum of each row is $1$
- A and B disjoint, A compact, and B closed implies there is positive distance between both sets
- Let the function $f: \to \mathbb R$ be Lipschitz. Show that $f$ maps a set of measure zero onto a set of measure zero
- Parametric Equations Problem
- Suppose $\mu$ is a finite measure on the Borel sets of $R$ such that $f(x) = \int_R f(x + t) \mu(dt)$ a.e., show $\mu(\{0\}) = 1$.
- Closure of a number field with respect to roots of a cubic