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In Functional Analysis, one often speaks of projective limits (limits in the category LCS of locally convex spaces) and inductive limits (colimits in LCS). In the latter case one has to be careful as colimits in TOP usually differ from those in LCS. What you state as a theorem says that every locally convex space is a projective limit of seminormed spaces. ...


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The weak topology has the property that the dual is the same as the dual with respect to the original topology, if $(E,\tau)$ is a topological vector space, and $E^\ast$ its (topological) dual, then $$(E,\sigma(E,E^\ast))^\ast = E^\ast.$$ For the topology of pointwise convergence on $C(S)$, the dual is easily described: if $\lambda \colon C(S) \to ...


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The first axiom for a norm space is N1: $\left\Vert \mathbf{x}\right\Vert =0\implies \mathbf{x}=\mathbf{0}$. Now, if we take this away and have the other two viz. homogenity and the triangle inequality, then we're left with what is called a seminorm. Any vector space over $\mathbb{R}$ or $\mathbb{C}$ can be converted into a seminorm space as follows: take ...


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I think your intuition is right: you already have the structure in place to talk about fuzzy tangent bundles (Definition 18). So you can talk about (fuzzy) sections of bundles, hence fuzzy vector fields, fuzzy forms and all that. So, if your question is "does the notion of fuzzy $C^1$ manifold you've defined lead to a natural notion of fuzzy differential ...


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Fuzzy logic determines the value of a Boolean variable. Using fuzzy logic we can say 'B is 30% false, 70% true'. Similarly a fuzzy point hovers in-between existence and non-existence. This means it has a range of values, and hence the (fuzzy) inverse does not exist, and so no (fuzzy) bijective mapping can exist, and so no (fuzzy) homeomorphism can exist.


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You can do it without any calculations: Let $T: X\to Y$ be a continuous linear map between two Banach spaces with dense range (e.g., $\ell^2\to\ell^2$, $(x_n)_n\mapsto (x_n/n)_n$ as in Theo's answer). Then take $Z=X\times Y$, $V_1=\lbrace (x,T(x)): x\in X\rbrace$ the graph of $T$ and $V_2= X\times \lbrace 0 \rbrace$. $V_1$ and $V_2$ are closed but their sum ...


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There's an answer here. I'm going to reproduce it here to fill in a couple of details. Let $$T : l^2 \rightarrow l^2 : (x_n) \mapsto \left(\frac{x_n}{n}\right).$$ Then $T$ is linear, and $$\|T(x_n)\|^2 = \left\|\left(\frac{x_n}{n}\right)\right\|^2 = \sum_{n=1}^\infty \frac{|x_n|^2}{n^2} \le \sum_{n=1}^\infty |x_n|^2 = \|(x_n)\|^2,$$ hence $T$ is bounded. ...


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We have$$||f||_1^n =\int_K |f^{(n)} (u)| du =\int_K 1\cdot |f^{(n)} (u)| du \leq \left(\int_K 1^2 du \right)^{\frac{1}{2}} \left(\int_K |f^{(n)} (u)|^2 du \right)^{\frac{1}{2}} =||f||^n_2$$ and $$f^{(n)} (t) =\int_0^t f^{(n+1)}(u) du \leqslant \int_K |f^{(n+1)}(u)| du =||f||^{n+1}_1$$ hence $$||f||^n_{\infty} \leq ||f||^{n+1}_1$$ and $$||f||^n_2 ...


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First, even though you don't say which book it is, I bet the book doesn't just say the $K_n$ are nested, I bet it assumes the stronger condition $$K_{n}\subset K_{n+1}^o,$$where $A^o$ is the interior of $A$. People sometimes says that compact sets satisfying this condition form an "exhaustion" of $\Omega$. If you assume that stronger condition then ...


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There is a positive result of Rolewicz for locally bounded F-spaces: S. Rolewicz, A generalization of the Mazur–Ulam theorem, Studia Math., 31 (1968), 501–505. There is an extension of this result due to W. Jian: W. Jian, On the Generalizations of the Mazur–Ulam Isometric Theorem, Journal of Mathematical Analysis and Applications, 263 (2001), ...


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Your proof is fine. In addition, looking at the proof of Th. 3.20 (c) of Papa Rudin's Functional Analysis, he just mentioned for metric spaces, closure of totally bounded sets are compact. Applying this to a compact set in a Frechet set, you conclude that the closed convex hull of a compact set is again compact. For the conclusion he uses the property (b) ...


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a topological vector space that is not a metric space: take $V=C(\Bbb{R})$ where the topology is given by convergence on compact sets. A basis for this topology is given by sets of the form $$U_{K,f,\varepsilon} = \{ g : \sup_K |g-f| < \varepsilon \}$$ where $f \in V$ is continuous, $\varepsilon >0$ is a positive real number, $K \subset \Bbb{R}$ is a ...



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