# Tagged Questions

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, as well as measure, ...

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### variational principle for the principal eigenvalue

I am reading the proof of theorem 2 in chapter 6 Evan PDE. I have difficulty verifying the following part of the proof, i.e. 3 questions here. 1) The assumptions $u\in H_0^1(U)$ and $u\in L^2(U)=1$ ...
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### Bounded linear extension of an operator whose co-domain is incomplete.

Let $X$ be a normed space and $Y$ be a Banach space. The bounded linear extension theorem states that a bounded linear operator $T : M \to Y$, where $M \subseteq X$ is dense, can be extended to a ...
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### Calculate the trace of $LBB^*$, where $L:H→H$ and $B:=ΦT^{1/2}$ for some $Φ:U_0→H$, an embedding $ι:U_0→V$ and $T:=ιι^*$

Let$^1$ $U$, $V$ and $H$ be $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with \langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\...
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### absolutely convergence in $l^2(\mathbb{N})$ space

Let $x=(x_i)_{i=1}^{\infty}$ be a sequence such that for all $y=(y_i)_{i=1}^{\infty}$ $\in l^2(\mathbb{N})$ $\sum\limits_{k=0}^{\infty}|x_iy_i|< \infty$ Show that $x \in l^2(\mathbb{N})$ . Can ...