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# Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \... 0answers 22 views ### Domain of definition of a hamiltonian with delta(contact) potential I am having a hard time making sense of the so-called "delta function potential well" in quantum theory. The Hamiltonian operator is defined as (with$\mathscr D_H\subset \mathscr H=L^2(\mathbb R)$) $$... 1answer 107 views ### L^2 bound and Sobolev spaces Let f \in L^2(\mathbb R) be a function such that$$\vert f \vert_{\alpha}:=\sup_{h>0}h^{-\alpha}\Vert f(\bullet+h)-f \Vert_{L^2}< \infty$$for some \alpha \in (0,1). I would like to know ... 0answers 48 views ### Are norm one projections in uniformly convex spaces unique? Let X be a uniformly convex Banach space and let M\subset X be a 1-complemented subspace. Is there a unique norm-one projection onto M or might there be more than one? In case the answer is ... 2answers 121 views ### Reference request on Min-Max theorem Consider the following min-max problem$$\inf_{x\in M} \sup_{y\in N} F(x,y),$$where F: M\times N\to\mathbb R is Lipschitz and y\mapsto F(x,y) is concave for all x\in M. Could we derive \... 0answers 90 views ### L^2 bound and interpolation of Hölder norm Consider the function$$F(x):=\int_{\mathbb R} f(t+x)f(t-x) \ dt .$$Clearly, we have by Cauchy-Schwarz$$\vert F(x) \vert\le \Vert f \Vert^2_{L^2} \vert F'(x)\vert\le 2\Vert f' \Vert_{L^2} \... 0answers 99 views ### What is the structure of a Banach space$X$when$Y$and$X/Y$are hereditarily indecomposable? Assume that$X$is a separable Banach space and$Y$a closed subspace such that$Y$and$X/Y$are hereditarily indecomposable (HI). The general question is what is the possible structure of$X$. ... 2answers 150 views ### Graph with complex eigenvalues The question I am wondering about is: Can the discrete Laplacian have complex eigenvalues on a graph? Clearly, there are two cases where it is obvious that this is impossible. 1.) The graph is ... 1answer 63 views ### Properties of the topology of sequential convergence$\tau_{seq}$Let$(X,\tau)$be a Hausdorff space. Denote by$\tau_{seq}$the topology on$X$whose closed sets are the sequentially$\tau$-closed subsets of$X$. I have read that$\tau_{seq}$has the following ... 1answer 52 views ### Is this kind of interpolation correct? Let$f=\sum f_j$be a finite sum. Assume that $$\|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$ $$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$ Then can we conclude that for$2<p<\infty$$$\|f\|_p\le C^{1-\... 0answers 25 views ### Numerical range of the first derivative operator on \{ u \in H^1(0,1): u(1)=0 \} I need to calculate the numerical range of the operator T:D(T)\subseteq L^2(0,1) \to L^2(0,1) defined by$$D(T):=\{ u \in H^1(0,1): u(1)=0 \}, \ Tu:=u', \ u \in D(T),$$where H^1(0,1) is the ... 1answer 77 views ### Regarding outer function being the quotient of two outer functions Let \mathbb{D} and \mathbb{T} denote the open unit disk and unit circle in \mathbb{C} respectively. We write Hol(\mathbb{D}) for the space of all holomorphic functions on \mathbb{D}. The ... 1answer 125 views ### Is there (fast) fourier transform for vector convolution? Given a list of variables u_1,\dots,u_m\in\mathbb R and v_1,\dots,v_n\in\mathbb R the standard convolution is defined$$U*V(t)={\sum_{i}} u_iv_{t-i}.$$Given a list of vectors u_1,\dots,u_m\in\... 1answer 95 views ### Significance and motivation for outer functions Let \mathbb{D} and \mathbb{T} denote the open unit disk and unit circle in \mathbb{C} respectively. We write Hol(\mathbb{D}) for the space of all holomorphic functions on \mathbb{D}. The ... 0answers 36 views ### Existence and uniqueness for semilinear problem Consider the following problem:$$-\Delta u + [(u)^+]^\alpha = 0,$$where$(\cdot)^+$is the positive part function and$\alpha >0\$. How does the theory of monotone operators provide existence ...

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