# Tag Info

• 52.4k
1 vote
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### Measurability of $\|f(\cdot, x_{2})\|_{L^\infty(X_{1})}$ (proof of Minkowski's inequality)

Thanks to @PhoemueX for the hint. We may use the $\sigma$-finiteness of $X_{1}$ to write $X_{1} = \bigcup_{n\geq 1}E_{n}$ with $\mu_{1}(E_{n}) < \infty$ and $E_{n}\subseteq E_{n+1}$ for all $n$. ...
• 1,157
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• 21.3k
1 vote
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### Vector space generated by translates

Your deduction on $E$ is correct, so I will answer question 3. It seems it is talking about the vector space of $L^2$ functions defined on some interval $I\subset \mathbb R^n$ with values in the ...
• 4,282

• 4,282
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### Confusion about Riesz representation theorem in $L^p$

For general $p$, this is far more difficult to construct than the representative in $L^2$ (which only needs the projection onto $\mathrm{ker}(T)$). In Alt - Linear Functional Analysis, this is done ...
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### Finding a function which is $L^1$ not $L^2$ and the integral is bounded by the square root.

We can simplify matters a bit. If the left side of the inequality is zero for some $A$, then the inequality holds trivially for such $A$. With this in mind, we can assume $f \equiv 0$ on $(-\infty, 0]$...
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• 2,521

### Rademacher functions form an orthonormal system but not an orthonormal basis

The system is complete but it's not a basis because it misses constant functions, i.e. $<1,r_n>=0$ for all n. In fact, the $L^2$ closure of the span of the $r_n$'s is also closed in the weaker ...
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### A property of solution operator of a elliptic PDE involving positive part of a function

I think this inequality cannot be true. Here is a counterexample for $N=1$: Define $u_n(x) := \sin(2n\pi x) \chi_{[0,1]}$. Then $u_n \rightharpoonup 0$ in $L^2(0,1)$ and in $L^2(\mathbb R)$. The ...
• 49.6k
1 vote
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### Showing $L^2$ - convergence of $\phi_n(X_n)$ to $\phi(X)$ when $\phi_n \to \phi$ and $X_n\to X$

This is not necessarily the case. Assume your probability space is the standard $((0,1), \mathcal L)$ ($\mathcal L$ is the lebesgue measure) and define the random variables $X_n$ as follows: X_n(x):=...
• 843

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