# Tagged Questions

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### Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...
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### Galerkin Orthogonality in this FEM?

Problem Galerkin orthogonality is but I am not sure if it is in the right form. How can you use this orthogonality here? I think I should expand the last inequality first somehow.
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### Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ ...
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### Prove that if $X$ is a Hilbert space, then $B(X)$ is not a Hilbert space

Im having a homework question that goes like this: X is a Hilbert space, a complete inner product space, show that B(X) is not a Hilbert space. Im quite stuck and I would love to understand this ...
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### I dont understand this notation

I`m having a homework question that goes like this: $X$ is a Hilbert space, a complete inner product space, show that $B(X)$ is not a Hilbert space. My only question for now is what does $B(X)$ ...
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### Fredholm operators in Hilbert spaces

Suppose $T_r$ and $T_l$ are the left and the right translations in $l_2$. $T_l$ maps $(x_1,x_2,x_3,...)$ to $(x_2,x_3,x_4,...)$, $T_r$ maps $(x_1,x_2,x_3,...)$ to $(0,x_1,x_2,...)$. It can be easily ...
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### Image and Kernel of a certain bounded operator

Consider and Hilbert Space $X$, $T\in B(X)$ and a scalar $\mu$ s.t. $|\mu|=||T||$ By a simple argument I deduced that $\ker(\mu I- T)=\ker(\bar\mu I-T^*$) where $*$ denotes the adjoint. I am then ...
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### Fourier series to calculate infinite series

I try to show that $\sum_{i=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ using Fourier series and $f(x) = x$ on $L^2_{\mathbb{C}}[-\pi, \pi]$, with basis $e_n(x) = \frac{1}{\sqrt{2\pi}}e^{inx}$. I ...
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### The Hilbert space $\mathcal{H}_\eta$ and unitary correspondence with $L^2[a,b]$

The question I have is related to a problem in Stein and Shakarchi's Real Analysis, Chapter 4. The problem Let $\eta(t)$ be a fixed strictly positive continuous function $[a,b]$. Define ...
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### Orthogonal Projectors

Please, I need help with this proble. Let $(H,\langle\cdot,\cdot\rangle)$ be a Hilbert space and let $V_1,V_2,\ldots,V_N$ closed subspaces, mutually orthogonal of $H$, that is, $v_i\perp v_j$ ...
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### Tight Probability on Hilbert space

I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$ And for any ...
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### Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$-\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0$$ on the bounded open set $U\subset\mathbb{R}^n$ ...
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### PDE weak solution problem

My professor grades really strictly (details). I would be very happy if you could help me with this problem: Let $U \subset R^n$ be a bounded set. Consider $\Delta^2 u = f$ on $U$ and ...
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### Particular series on Hilbert Space

Let $(H, \langle\cdot,\cdot\rangle)$ a Hilbert space and consider a sequence $\{x_n\}_{n\in\mathbb{N}}$ of $H$ such that: \langle x_n,x_m\rangle\ =\ \delta_{mn}\ =\ \left\{\begin{array}{ll}1, & ...
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### Bounded operator and Compactness problem

Let $H$ be a Hilbert space with orthonormal basis $(e_{n})_{n\in\mathbb{N}}$. Furthermore, let $T\colon H\rightarrow C[a,b]$ be a bounded operator. a) Let $x\in [a,b]$. Show that there is a ...
I want to show that if linear mapping $L:B_1\rightarrow B_2$ is isomorphism of Banach space and $\|L(x)\|_{B_1} =\|x\|_{B_2}$ (surjective and isometry) so it consist that $L$ is isomorphism of ...
Let $H$ be a Hilbert space and that $X$ are bounded. Suppose $X$ is self-adjoint. Show that $Y=X+iI$ is invertible and the inverse $Y^{-1}$ has the norm $\lVert Y^{-1} \rVert \le 1$. I can prove $Y$ ...