# Tagged Questions

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### 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 ...
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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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### Operator norm estimate

Let $H$ be a Hilbert space with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$. Furthermore, let $B\colon H\rightarrow C[a,b]$ be a bounded operator. According to the Riesz-Frechet theorem there is ...
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### Norm of a function, Smoothness Penalization

I am seeking for some intuition why norm (for any reasonable norm on functions) of a function is smaller if the function is smoother.
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### Invertible operator norm bound

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 $Y$ ...
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### Operator norm and spectrum

Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$? ...
How to prove that in a Hilbert space $H$, $$\lVert h \rVert = \sup_{u \in H}\frac{|(h,u)|}{\lVert u \rVert}?$$ Showing that the RHS is $\leq$ the LHS is easy but not sure of the other part. This is ...