# Tagged Questions

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

1answer
47 views

### Weak Solutions to PDES

I am working through some practice problems for my PDE class and I came across the following: Let $U\in \mathbb{R}^n$ be a smooth, bounded, connected open set. Let $\Gamma_1$, $\Gamma_2$, be two ...
3answers
47 views

### Two operators $X$ and $Z$ in an infinite dimensional Hilbert space satisfying $X^2=Z^2=I$ and $\{X,Z\}= 0$

I am seeking to extend the following theorem to the case of infinite dimensional Hilbert space: Suppose we have two Hermitian operators $X$ and $Z$ in a finite dimensional Hilbert space $\mathcal H$. ...
1answer
43 views

### Projection Theorem

I've been trying to apply the projection theorem to the following problem with no success. I've spent a few hours on this today, any help would be appreciated. Let H be a finite dimensional Hibert ...
3answers
103 views

### Natural ways in which the *complex* valued L-integral and *complex* Hilbert spaces come up

I have two questions regarding how two concepts that involve complex valued functions may come up in a natural way. (Non-natural ways are: These concept come up in order to present a unified theory, ...
0answers
45 views

### How can we use theory from $L^2(\mathbb{R})$ on a sequence of numbers (discrete signal)

In have problems understanding connection between theory that is done in $L^2(\mathbb{R})$ and its application on discrete signal. look at this paper http://home.ustc.edu.cn/~zhanghan/cs/...
1answer
57 views

### Show that $\langle y_i, y_j\rangle = 0 \forall i \neq j.$

Let $y_1, y_2, . . .$ be a sequence in a Hilbert space. Let $V_n$ be the linear span of $\{y_1, y_2, . . . , y_n\}$. Assume that for $n \ge 1, ∥y_{n+1}∥ \le ∥y − y_{n+1}∥$ for each $y \in V_n$. ...
1answer
36 views

1answer
57 views

0answers
36 views

### If $Q$ is an operator on a Hilbert space $U$, $U_0:=Q^{1/2}(U)$ and $(e_n)_n$ is a basis of $U_0$, then $u↦\sum_na_n(u,e_n)_0e_n$ is an embedding

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space and $\left\|\;\cdot\;\right\|$ be the norm induced by $\langle\;\cdot\;,\;\cdot\;\rangle$ $Q$ be a bounded, linear, ...
0answers
39 views