# The decomposition of inner product space

If $V$,$W$are two inner product spaces and $L:V\to W$ is a linear map with its adjoint $L^\star$, then is there a decomposition of $W$=ker$(L^\star)$ $\oplus$ im$(L)$ ? (It is easy that the conclusion holds if $V$ and $W$ are finite-dimensional)

-

What if $\operatorname{im}{L}$ isn't closed? – t.b. Apr 4 '12 at 9:57
Which duals do you consider? Algebraic or continuous? – Norbert Apr 4 '12 at 12:13
@Norbert algebraic, since we do not assume that $L$ is continuous. – Hezudao Apr 4 '12 at 13:17
@t.b. I see. If the decomposition does exist, then im($L$) should be closed. But what if we assume first that im($L$) is closed? – Hezudao Apr 4 '12 at 13:42
I think, $L^*$ is not well definied, so you should give your definition. – Norbert Apr 6 '12 at 5:58
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9123654365539551, "perplexity_flag": "middle"}