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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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Why locally compact hausdorff space

I may be missing something very trivial,but cannot figure it out. To prove the fact that the space $C_0(X)$ is a Banach space under the usual sup norm when $X$ is locally compact and Hausdorff, where do we use the properties of $X$? Thanks for any…
Ester
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Why in uniformly convex Banach space every non empty, closed, convex subset contains a unique element of smallest norm?

In Hilbert space every non empty, closed, convex subset contains a unique element of smallest norm. Why is that true also in Banach space which is uniformly convex? (normed space which is uniformly convex is a space in which for all sequences…
user18217
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Does the open mapping theorem have a local version?

Let $T:X\to Y$ be a linear continuous surjection between Banach spaces $X$ and $Y$. By the open mapping theorem, we have $T$ is open. Now let $C$ be a closed convex subset of $X$ satisfying that $T(C)$ is also closed. Does $T:C\to T(C)$ is open?
luozhenghua
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Complexification of a real Hilbert space

The following is problem 1.7 in chapter 1 of Conway's A Course in Functional Analysis. Let $H$ be a Hilbert space over $\Bbb R$ and show that there is a Hilbert space $K$ over $\Bbb C$ and a map $U: H \to K$ such that a) $U$ is linear b)…
saeed
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If a linear functional is not bounded, then it has a non-closed kernel.

In this answer, the OP gave a proof to the proposition that if a linear functional $f$ is not bounded, then it has a non-closed kernel. However, he only showed that if $f$ is not bounded, then a unbounded sequence exists in the kernel, which doesn't…
xzhu
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To show that the complement of the kernel of an unbounded linear functional is path connected

The kernel of a linear functional is either closed (if it is continuous) or dense (if not). In the latter case, apparently the complement of the kernel is path connected. How does one see this? Yes Daniel, this is the point. The question was on…
Paul
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Spectrum of the multiplication operator

Let $B[0,1]$ be the Banach space of bounded complex functions on $[0,1]$ endowed with the supremum norm. I've have to show that the spectrum of the multiplication operator $T_q: B[0,1] \rightarrow B[0,1]$ $$ (T_q f)(t) : = q(t)f(t), \,\,\, t \in…
simon
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Compact subset of a Banach space of infinite dimension

Let $X$ be a Banach space of infinite dimension. And let $K\subset X$ be a compact subset of $X$. Can we conclude something about the interior of $K$? Is it true that it's empty? I don't know how to attack this problem. I have not even examples of…
Shanks
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Construct dense subspace of codimension $n$ for all $n$

I want to prove the following: Let $X$ be an infinite dimensional normed space. For all integer $n\geq1$: $X$ has a dense subspace of codimension $n$, i.e. a subspace $L$ such that $\dim(X/L)=n$. How can i do this? My first thought was: Take a basis…
user95344
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Is exponentiation defined for non-self-adjoint operators?

In a book I am reading (Blank, Exner, Havlíček: Hilbert Space Operators in Quantum Mechanics), functions of operators are defined via spectral decomposition for self-adjoint operators. Spectral decomposition is also well-defined for unitary…
The Vee
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What is the closure of the Laplacian on $L^2(0, \infty)$ with domain $D(\Delta):=C^\infty_0(0, \infty)$?

Let $\Delta$ be the operator on $L^2(0, \infty)$ defined as follows: $\Delta \phi:= \phi''$, with domain $D(\Delta):=C^\infty_0(0, \infty)$. Is $\Delta$ closed or closable? In the case, what is its closure?
BGA
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Extension of equivalent norm in subspace to the whole space

I'm trying to prove the following result: Let $(X, \|\,\cdot\,\|_X)$ be a normed space and $Y$ a subspace of $X$. If $\|\,\cdot\,\|_{YY}$ is an equivalent norm to $\|\,\cdot\,\|_Y$ (the inherited norm from $X$) in $Y$, then there exists a norm…
Eparoh
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Are there corner cases when using Hilbert basis and bounded operators?

It seems that with a Hilbert basis and a bounded operator, you can do all the calculations while ignoring convergence issues, as if you were in a finite dimensional case. For example if $a_{ij}$ are the coefficients of operator $A$ in the basis,…
user2370139
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Explicit form of the homeomorphism between $C[0,1]$ and $C[0,1]\setminus 0$

How to construct the homeomorphism between $C[0,1]$ and $C[0,1]\setminus\{\theta\}$ in the explicit form? Here, as usual, $C[0,1]$ is the Banach space of continuous functions $f:[0,1]\to\mathbb{R}$ with the…
Ilnara
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