# Tagged Questions

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

69 views

### How do you prove a hilbert transform?

I am stuck with this question below, I need help;
48 views

### Compact Operator on Hilbert Space

How do I show that the range of $\lambda I-T$ is all of $H$ (Hilbert Space) if and only if the null-space $\bar\lambda I-T^{\ast}$ is trivial? Thanks!
19 views

152 views

### Operator norm and Hilbert Schmidt norm

I'm looking for a proof of $$||T||\leq ||T||_{HS},$$ for which it is sufficient to show ||Tx|| \leq ||x|| \cdot ||T||_{HS} \forall x\in H, x\not=0 \end{...
39 views

### Prove that the space P is a Hilbert Space.

Prove that the space P of all entire functions of the form $$f(z)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi(t) e^{-izt} dt,$$ is a Hilbert Space, where $\varphi\in L^2[-\pi,\pi]$. The inner product of ...
113 views

67 views

### How to show that $\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$

Show that $$\|a+b+c\|^2\leq 3\|a\|^2+3\|b\|^2+3\|c\|^2$$ where $a,b,c$ are in some Hilbert space $(H,\langle\cdot,\cdot \rangle)$? I see that we have $$\|a+b\|^2\leq2\|a\|^2 +2 \|b\|^2$$ due to the ...
48 views

38 views

### Hilbert subspaces of $B(\mathbb{R}^n)$

Apart from the one-dimensional subspaces, what are the Hilbert subspaces of $B(\mathbb{R}^n)$? I'm not even sure if such subspaces exist.