If the spaces are Banach then the answer is No. Look at the closed graph theorem.
Dear Sudhanshu Aggarwal ,
If f(x) : X → Y is linear map and X has finite dimension,
then f should be a continuous map.
If we consider the set X of all differentiable maps defined over [0,1],
define T : X → X , T(f) = df/dx(0), it is a linear discontinuous map an it is
a closed operator, that is, it generates closed graphs.
See
(1)https://en.wikipedia.org/wiki/Discontinuous_linear_map
(2) https://en.wikipedia.org/wiki/Unbounded_operator
.
Best regards
There are so many situations in which the answer is NO, because of the closed graph theorem (such as in Fréchet spaces). However, if E is a Banach space, then the identity I from (E, \sigma) into E has closed graph, but is continuous only if E is finite dimensional, here \sigma denotes the weak topology $\sigma(E, E')$.
no, a discontinuous linear function generally does not have closed graph.
Dear Nidhi,
A continuous linear map has closed graph. But a discontinuius linear map may also have a closed graph. This cannot happen between Banach spaces, but there are other topological vector spaces that fail to be Banach. I gave an example in my previous answer.
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