# Tagged Questions

For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

11 views

52 views

### In an orthogonal sequence is weakly convergent, then it is convergent

Let $H$ be a Hilbert space, and let $\{x_k \}_{k\in \mathbb{N}}$ be an orthogonal subset of $H$. If for every $y\in H$, $\sum \left<x_k, y\right>$ converges, then $\sum x_k$ converges ...
13 views

16 views

21 views

### If $f_n \rightharpoonup f$ and $g_n \rightharpoonup g$, and $|f_n|_H - |g_n|_H \to |f|_H-|g|_H$, does $f_n \to f$ and $g_n \to g$?

We work in a Hilbert space $H$. If $f_n \rightharpoonup f$ and $g_n \rightharpoonup g$, and $$|f_n|_H - |g_n|_H \to |f|_H-|g|_H$$ is there any chance that $f_n \to f$ and $g_n \to g$? Of course this ...
32 views

### Find a function $b$ such that the operator $\frac{d}{dx}+b(x)$ is symmetric with the weight $x^2$

Find the value of $b(x) \in \mathbb{C}, x\in \mathbb{R}$, so that $$Â=(Â^{*})^{t}$$ with $$Â=i\frac{d}{dx}+b(x)$$ Here, $(f|g)$ is defined by $$\int_{-\infty}^{\infty} x^{2}f^{*}(x)g(x)dx$$ I ...