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Questions tagged [functional-analysis]

Functional analysis, 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 and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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A question about Duality in Hilbert Spaces

I'm confused at the construction of the Riesz Representation for a Hilbert space. Given a Hilbert space $H$, we define $J(y) := ( * \space | \space y) \in H'$ This is an element of the dual space $H'$. My confusion that I want to clarify is this: on…
JJJ
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Locally convex versus uniformly convex

This is to get some clarity, I know the two words have nothing to do with each other but they sound so similar that I caught myself saying "locally convex" when I really meant "uniformly convex" and the other way around. Perhaps it would help me if…
Rudy the Reindeer
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The finite intersection of absorbing set is absorbing

I got stuck proving the finite intersection of absorbing set is absorbing. Can anyone help?
89085731
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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 you please give some hint about solving this problem.…
Serkan Yaray
  • 473
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Unbounded operators in Hilbert space.

If $A\subset B $, $A$ not equals to $B$, are unbounded operators, I need to prove that: $(a)$ If $B$ is selfadjoint, then $A$ is symmetric, but not selfadjoint. $(b)$ If $A$ is selfadjoint, then $B$ is not symmetric.
Nina
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The remark of closed graph theorem in Rudin's book

In Rudin's functional analysis, the remark of closed graph theorem says the statement: If $\{x_n\}$ is a sequence in $X$ such that $x=\displaystyle\lim_{n\rightarrow\infty} x_n$, $y=\displaystyle\lim_{n\rightarrow\infty}f(x_n)$ exist, then…
89085731
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Operator Convergence in Hilbert space

I'm wondering if you can check my work and provide help on part of this problem. Here's the question: Let $H$ be Hilbert and $(\phi_n)$ an orthonormal basis. Let $(a_j)$ be a sequence of nonnegative reals such that $a_j \to \infty$. Define $T_n(t)…
user288742
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Norm of operator in $\mathcal{L}(l_2,l_2)$

Let $(a_n)\in l_\infty$ and $ T:l_2 \to l_2$ given by $T((x_n)) = (a_nx_n)$. Show that $T \in \mathcal{L}(l_2,l_2)$ $\|T\| = \|(a_n)\|_{\infty}$ It's easy to prove that $T$ is continuous since $T$ is linear and: $$\|T((x_n))\|_2 =…
user344460
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Why is this true $v \in H_2(\Omega) \Rightarrow v \in C(\Omega)$?

As the title say, I'm trying to figure out why is the following true: $$v \in H_2(\Omega) \Rightarrow v \in C(\Omega)$$ with $H_2(\Omega) = \{f: \Omega \to \mathbb{R}: \|f^{(k)}\|_{L_2} < \infty, \forall 0 \leq k \leq 2\}$ (say $\Omega = [0,1]$). I…
Shawn
  • 123
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Uniform continuity in $\mathbb{R}^n$

I have trouble with this question: What does it mean that a function is uniformly continuous in a domain $D\subset\mathbb{R}^n$ ? I know the definition for uniform continuity , but im not sure if this is enough. Since our domain can be a subset of…
Biggiez
  • 159
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A question about function $f(x)=e^{-x} \sqrt{x}$ and $g(x)=\frac{x^2-2ax+1}{x+1}$.

Let $f(x)=e^{-x} \sqrt{x}$ , $g(x)=\frac{x^2-2ax+1}{x+1}$ and $a>\ln 2-1$. Prove $f(x)g(x)<\frac{1}{2}$. This problem is one of my homework. After my studying, I know $f(x)_{max}=\frac{\sqrt{2e}}{2e}<\frac{1}{2}$.
C. Simon
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Identity operator on a dense subset cannot be extended to a continuous function from X to Y

I'm practicing some old exams for my Functional Analysis test and i'm stuck on the following question: Let $X$ be any Banach space with $Y \subset X$, $Y \neq X$ and $Y$ dense in $X$. Show that the identity operator on $Y$ cannot be extended to a…
ronalddb89
  • 141
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convex set containing origin which is not balanced in TVS

In topological vector space, intuitively a balanced set is a set which doesn't have a hole. Suppose $V$ is a convex set containing origin, can anyone give a counterexample which is not balanced?
89085731
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Is there an easy example of a strictly-convex space that is not uniformly convex?

I've found on the web that $\ell^1$ equipped with the norm $\|\cdot \|=\|\cdot\|_1+\|\cdot\|_2$ is an example. However, I can't prove this to see that it is true. Can someone show me how, or provide a different, and preferably, easier example.…
user337578
  • 35
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A unit closed ball in a n.v.s is totally bounded?

Let $X$ be an infinite-dimensional normed linear space. $\overline{B}(0,1):=\{x\in X:||x||\leq 1\}$. My question is, $\overline{B}(0,1)$ is totally bounded? If $X$ is complete, that is to say, $X$ is a Banach space, then $\overline{B}(0,1)$ is…
James Chan
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