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It is said that functional analysis is just infinite dimensional version of linear algebra. However, I am quite puzzled by this statement since we are mainly doing analysis on it. Another question is that why we want to define topology on vector space, is it because we can let the function to be continuous so that we can gain some benefit from it, e.x. the continuous map of a compact set is bounded. In other words, what we really care is just mapping itself regardless of the topology

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One way to understand why we'd want a topology, especially a metric, or a norm, or an inner product, on an infinite-dimensional vector space ... is that the infinite-dimensional vector space is a naturally-occurring space of functions of a particular sort, perhaps functions on an interval $[a,b]$ on the real line. For use, we'd want to be able to take limits (in that metric or topology) and be sure to stay in the same class of functions. For example, a good norm on continuous functions on $[a,b]$ is $|f|=\sup_{x\in[a,b]}|f(x)|$. It is a standard exercise that a Cauchy sequence of continuous functions (with respect to the metric $d(f,g)=|f-g|$) has a limit which is also a continuous function.

Similarly, but often more subtly, vector spaces of other types of functions can often be given metrics or other topologies so that they are suitably complete, or, oppositely, we find reason to take the completion with respect to a "good" metric. An example of a very good metric is that coming from an inner product, such as $\langle f,g\rangle=\int_a^b f(x)\overline{g(x)}\,dx$ for functions on that interval. If we start with continuous functions, the space is not complete, but we can take the completion, to get the Hilbert space $L^2[a,b]$. Linear operators on a Hilbert space have spectral theory strongly resembling that in finite-dimensional linear algebra, but, unsurprisingly, with some complications. But, in many useful situations, there are orthonormal bases consisting of eigenvectors/eigenfunctions for a given (nice...) operator.

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  • $\begingroup$ Thanks for your answer. What I mean is that if we define the topology to serve the mapping or not?e.x. To make it continuous? $\endgroup$ – 89085731 Jul 8 '17 at 23:04
  • $\begingroup$ Like we restrict the range of the space for fixed point theorem $\endgroup$ – 89085731 Jul 8 '17 at 23:05
  • $\begingroup$ The true answer is "it depends". With various natural mappings among spaces of functions, sometimes we topologize one or more with the exact purpose to make the mapping continuous, yes. Sometimes we want a certain topology on the vector space, and there is some conflict with continuity of operators ("unbounded operators" such as $d/dx$ on Hilbert spaces such as $L^2[a,b]$) and the operators may not even be definable on the whole space...Many possibilities. $\endgroup$ – paul garrett Jul 8 '17 at 23:21
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When passing to infinitely many dimensions, topological (hence analytical) notions become much more important. According to a classical theorem in introductory functional analysis, every finite dimensional topological vector space over the complex field is linearly homeomorphic to $\mathbb{C}^n$, where $n$ is the dimension of the vector space in question. Afterwards, it is usually proven that a linear mapping from $\mathbb{C}^n$ to any topological vector space is always bounded, hence continuous.

Therefore, if you consider only finite dimensional spaces, you gain nothing by assuming continuity. In the infinite dimensional case, however, there exists many discontinuous linear mappings, especially linear functionals. The study of conditions to ensure a sufficiently large supply of bounded linear functionals leads to the notion of local convexity and the theorems of Hahn-Banach. Another example is compactness of linear operators: if we restrict ourselves to finitely many dimensions, every linear mapping is in fact compact. In spectral theory, one discovers that compactness is the key to the desirable spectral properties of matrices known from linear algebra. In my opinion, this fact is why people often think of functional analysis as infinite dimensional linear algebra.

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Topology gives a notion of convergence and approximation. In this way techniques of finite-dimensional vector spaces can be brought to bear on the infinite-dimensional, provided there is some way to approximate elements by finite linear combinations of some set of vectors. And linear mappings can be studied through approximation, provided the linear mappings are continuous with respect to the topology. Dual space elements can be studied if one restricts to continuous linear functionals. One of the earliest developments in this regard was the characterization of a continuous linear functional on $C[0,1]$ as a Riemann-Stieltjes integral with respect to a function of bounded variation, which Riesz discovered in the first decade of the 20th century, at least a decade before a definition of a general topology.

Notions of compactness on function spaces provided a useful and important means by which to understand Fredholm's original work on integral equations, and that was the motivation behind the definition of a compact operator, even before notions of General Topology had been developed.

Topology and continuuity provide approximation and convergence, and that's why topology is so critical in studying infinite-dimensional spaces.

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