# Tagged Questions

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

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### $P+Q-PQ$ is a projection if and only if $PQ=QP$.

Let $\mathcal H$ is a Hilbert space and $P,Q:\mathcal H \to \mathcal H$ are projections. I want to show that $P+Q-PQ$ is a projection if and only if $PQ=QP$. If $PQ=QP$ clearly $P+Q-PQ$ is a ...
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### From the direct sum theorem, how can we deduce that $y$ is the orthogonal projection of $x$?

In the direct sum theorem we have $$H =Y \oplus Z$$ where $Y$ is any closed subspace of a Hilbert space $H$. It is easy to deduce that for every $x \in H$ there is a $y \in Y$ such that $$x=y+z$$. ...
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### In what sense are compact operators limits of finite-rank operators?

The convergence is in respect to what topology ? Can someone please write it mathematically ?
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### Is there any connection between these two definitions of coercivity (ellipticity) in PDE and bilinear form?

In the field of partial differential equation, we say that the following operator $$Lu=-\sum\limits_{i,j=1}^n (a^{ij}u_{x_i})_{x_j}+\sum\limits_{i=1}^n b^i u_{x_i}+cu$$ is (...
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### Strong convergence on the unit sphere of $l_2$

Let $(p_n)$ be a strictly increasing sequence of natural numbers and $(\epsilon_n)$ a positive sequence decreasing to $0$. Suppose $x_n$ is a sequence in $S(l_2)$ (the unit sphere of $l_2$) with the ...
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### How to calculate the norm of this operator?

Let $H$ be a separable Hilbert space and $(\phi_k)$ be a basis $A(t)$ is defined such as $A\phi_k=\exp(-t/k)\phi_k$. I am specifically interrested whether $\|A(t)\| \to 0$ when $t \to \infty$ or not, ...
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### Is the set $\ell ^2$ a $G_\delta$ in $\mathbb R ^\omega$?

$\ell ^2$ = set of sequences $(x_i)$ of real numbers such that $\|x\|=\sum _{i=0} ^\infty x_i ^2<\infty$. Question: Is $\ell ^2$ a $G_\delta$-set in the product topology $\mathbb R ^\omega$? I am ...
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### Unitary space: prove that

How I can start this problem? $X$ is unitary space. Prove that if $M_1, M_2 \subset X:$ $M_1\neq \emptyset ,M_2\neq \emptyset$ and $M_1 \subset M_2$ then $M_2^\perp \subset M_1^\perp$ Thank ...
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### Do powers of contraction on Hilbert space converging to zero imply convergence of its adjoint to zero also?

In my functional analysis class I was met with the following problem: We suppose that $\mathbb{H}$ is a Hilbert space and that T is a contraction operator on H (meaning $||T|| \leq 1$ in the ...
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### Generalized Mixed Models and Hilbert Spaces

I recently came across the derivation of the normal equations for linear regression using Hilbert Spaces and projection theorem, and thought it was pretty cool. After doing a lot of googling, it seems ...
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### Are those two definitions of orthogonal projection equivalent in a general Hilbert space?

I am taking a graduate level course in probability and we started off with some results in functional analysis. One thing that I feel I do not understand properly is the definition of an orthogonal ...
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### $A$ and $A^*$ dissipative implies $D(A) \subset H$ is compact embedding

For selfstudy purpose I want to show the following: $H$ Hilbertspace, $D(A)$ dense subspace of $H$, $A\colon H \supset D(A) \to H$ linear closed dense defined operator. If $A$ and $A^*$ are both ...
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### Complete orthonormal system in a finite dimension Hilbert space

I have to solve the following problem of functional analysis. Let $H$ be a Hilbert space of dimension $N$. Prove that every complete orthonormal system in $H$ has $N$ elements and that $H$ is ...
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### Is there a relation between Cartesian and tensor product of function spaces and function factorizability

H1 and H2 are two Hilbert spaces represented by a function space, say f1(x1) and f2(x2) are its vectors. If H3 is tensor product of H1 and H2 I assume one can say that f(x1,x2) now represents vectors ...
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### If the sum of two weakly convergent sequences strongly converges, do the summands strongly converge?

Let $a_n \rightharpoonup a$ and $b_n \rightharpoonup b$ weakly in $H$, a Hilbert space, and suppose that $a_n + b_n \to a+b$ strongly in $H$. Is it true that $a_n \to a$ and $b_n \to b$ strongly? I ...
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### What's the value of $\alpha$ satisfying $||f'||^2\ge \alpha||f||^2$? [duplicate]

I am reading a paper about numerical analysis of a certain method for solving operator equation. Let our Hilbert space be $L^2[0,1]$, we define the subspace $D\in L^2[0,1]$ by  D:=\{f\in C^2(0,1)\...
In one dimension, the space $L^{2}([0,1], dx)$ of complex-valued square integrable functions on $[0,1]$ is well known to be separable, and sine and cosine provide an explicit Hilbert basis for it. My ...